From ee72175941ba27b330519b418d9714709b1aa6e9 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Fri, 16 Oct 2015 21:42:07 +0000 Subject: Initial commit --- BBlog.sty | 46 + Figs/3x2_skel.pdf | Bin 0 -> 1783 bytes Figs/3x2_subskel.pdf | Bin 0 -> 1637 bytes Figs/Lshape.pdf | Bin 0 -> 1501 bytes Figs/Lshape_label.pdf | Bin 0 -> 11372 bytes Figs/bethe.pdf | Bin 0 -> 3573 bytes Figs/boundary_circuit.pdf | Bin 0 -> 2338 bytes Figs/boundary_circuit_label.pdf | Bin 0 -> 13842 bytes Figs/boundary_circuit_naked.pdf | Bin 0 -> 13128 bytes Figs/circuit.pdf | Bin 0 -> 1573 bytes Figs/comb.pdf | Bin 0 -> 5140 bytes Figs/enclosed_example.pdf | Bin 0 -> 2068 bytes Figs/enclosed_example_auxiliary_cover.pdf | Bin 0 -> 21791 bytes Figs/enclosed_example_cover.pdf | Bin 0 -> 21279 bytes Figs/enclosed_example_directed.pdf | Bin 0 -> 11956 bytes Figs/enclosed_example_monomer_auxiliary_cover.pdf | Bin 0 -> 21272 bytes Figs/enclosed_example_monomer_cover.pdf | Bin 0 -> 20765 bytes Figs/etareplace_cross.pdf | Bin 0 -> 1139 bytes Figs/etareplace_empty.pdf | Bin 0 -> 1329 bytes Figs/etareplace_occupied.pdf | Bin 0 -> 1330 bytes Figs/etareplace_r.pdf | Bin 0 -> 1322 bytes Figs/even_example.pdf | Bin 0 -> 1967 bytes Figs/even_example_auxcover.pdf | Bin 0 -> 12487 bytes Figs/even_example_auxiliary.pdf | Bin 0 -> 12442 bytes Figs/even_example_cover.pdf | Bin 0 -> 12786 bytes Figs/even_example_label.pdf | Bin 0 -> 12753 bytes Figs/even_example_label_pre.pdf | Bin 0 -> 12444 bytes Figs/even_example_monomer.pdf | Bin 0 -> 12323 bytes Figs/extended_dimer.pdf | Bin 0 -> 19552 bytes Figs/extended_dimer_triangle.pdf | Bin 0 -> 22197 bytes Figs/minimal_circuit.pdf | Bin 0 -> 1414 bytes Figs/noncoverable.pdf | Bin 0 -> 1730 bytes Figs/oddrepl_aft.pdf | Bin 0 -> 1396 bytes Figs/oddrepl_bef.pdf | Bin 0 -> 1220 bytes Figs/simple_example.pdf | Bin 0 -> 1571 bytes Figs/simple_example_auxiliary.pdf | Bin 0 -> 11040 bytes Figs/simple_example_labels.pdf | Bin 0 -> 11641 bytes Figs/simple_example_monomers.pdf | Bin 0 -> 10964 bytes Figs/triangle_3.pdf | Bin 0 -> 1309 bytes Figs/triangle_3_Xcover1.pdf | Bin 0 -> 1140 bytes Figs/triangle_3_Xcover3.pdf | Bin 0 -> 1140 bytes Figs/triangle_3_cover1.pdf | Bin 0 -> 1327 bytes Figs/triangle_3_cover3.pdf | Bin 0 -> 1332 bytes Figs/triangle_3_pre.pdf | Bin 0 -> 1133 bytes Figs/triangle_4.pdf | Bin 0 -> 1442 bytes Figs/triangle_4_pre.pdf | Bin 0 -> 1124 bytes Figs/triangle_5.pdf | Bin 0 -> 1608 bytes Figs/triangle_5_directed.pdf | Bin 0 -> 10716 bytes Figs/triangle_5_pre.pdf | Bin 0 -> 1145 bytes Figs/triangle_groups_l.pdf | Bin 0 -> 32303 bytes Figs/triangle_groups_lr.pdf | Bin 0 -> 33548 bytes Figs/triangle_groups_r.pdf | Bin 0 -> 10718 bytes Giuliani_Jauslin_Lieb_2015.tex | 1642 +++++++++++++++++++++ README | 53 + bibliography.BBlog.tex | 33 + header.sty | 15 + iansecs.sty | 509 +++++++ kiss.cls | 47 + toolbox.sty | 46 + 59 files changed, 2391 insertions(+) create mode 100644 BBlog.sty create mode 100644 Figs/3x2_skel.pdf create mode 100644 Figs/3x2_subskel.pdf create mode 100644 Figs/Lshape.pdf create mode 100644 Figs/Lshape_label.pdf create mode 100644 Figs/bethe.pdf create mode 100644 Figs/boundary_circuit.pdf create mode 100644 Figs/boundary_circuit_label.pdf create mode 100644 Figs/boundary_circuit_naked.pdf create mode 100644 Figs/circuit.pdf create mode 100644 Figs/comb.pdf create mode 100644 Figs/enclosed_example.pdf create mode 100644 Figs/enclosed_example_auxiliary_cover.pdf create mode 100644 Figs/enclosed_example_cover.pdf create mode 100644 Figs/enclosed_example_directed.pdf create mode 100644 Figs/enclosed_example_monomer_auxiliary_cover.pdf create mode 100644 Figs/enclosed_example_monomer_cover.pdf create mode 100644 Figs/etareplace_cross.pdf create mode 100644 Figs/etareplace_empty.pdf create mode 100644 Figs/etareplace_occupied.pdf create mode 100644 Figs/etareplace_r.pdf create mode 100644 Figs/even_example.pdf create mode 100644 Figs/even_example_auxcover.pdf create mode 100644 Figs/even_example_auxiliary.pdf create mode 100644 Figs/even_example_cover.pdf create mode 100644 Figs/even_example_label.pdf create mode 100644 Figs/even_example_label_pre.pdf create mode 100644 Figs/even_example_monomer.pdf create mode 100644 Figs/extended_dimer.pdf create mode 100644 Figs/extended_dimer_triangle.pdf create mode 100644 Figs/minimal_circuit.pdf create mode 100644 Figs/noncoverable.pdf create mode 100644 Figs/oddrepl_aft.pdf create mode 100644 Figs/oddrepl_bef.pdf create mode 100644 Figs/simple_example.pdf create mode 100644 Figs/simple_example_auxiliary.pdf create mode 100644 Figs/simple_example_labels.pdf create mode 100644 Figs/simple_example_monomers.pdf create mode 100644 Figs/triangle_3.pdf create mode 100644 Figs/triangle_3_Xcover1.pdf create mode 100644 Figs/triangle_3_Xcover3.pdf create mode 100644 Figs/triangle_3_cover1.pdf create mode 100644 Figs/triangle_3_cover3.pdf create mode 100644 Figs/triangle_3_pre.pdf create mode 100644 Figs/triangle_4.pdf create mode 100644 Figs/triangle_4_pre.pdf create mode 100644 Figs/triangle_5.pdf create mode 100644 Figs/triangle_5_directed.pdf create mode 100644 Figs/triangle_5_pre.pdf create mode 100644 Figs/triangle_groups_l.pdf create mode 100644 Figs/triangle_groups_lr.pdf create mode 100644 Figs/triangle_groups_r.pdf create mode 100644 Giuliani_Jauslin_Lieb_2015.tex create mode 100644 README create mode 100644 bibliography.BBlog.tex create mode 100644 header.sty create mode 100644 iansecs.sty create mode 100644 kiss.cls create mode 100644 toolbox.sty diff --git a/BBlog.sty b/BBlog.sty new file mode 100644 index 0000000..5448d80 --- /dev/null +++ b/BBlog.sty @@ -0,0 +1,46 @@ +%% +%% BBlog bibliography related commands +%% + +%% length used to display the bibliography +\newlength{\rw} +\setlength{\rw}{1.75cm} + +%% read header +\IfFileExists{header.BBlog.tex}{\input{header.BBlog}}{} + +%% cite a reference +\def\cite#1{% +\ref{cite#1}% +%% add entry to citelist after checking it has not already been added +\ifcsname if#1cited\endcsname% +\expandafter\if\csname if#1cited\endcsname% +\else% +\csname if#1citedtrue\endcsname% +\immediate\write\@auxout{\noexpand\BBlogcite{#1}}% +\fi% +\else% +\expandafter\newif\csname if#1cited\endcsname% +\csname if#1citedtrue\endcsname% +\immediate\write\@auxout{\noexpand\BBlogcite{#1}}% +\fi% +} +%% an empty definition for the aux file +\def\BBlogcite#1{} + +%% an entry +\long\def\BBlogentry#1#2#3{ + \hrefanchor + \outdef{cite#1}{#2} + \parbox[t]{\rw}{[\cite{#1}]}\parbox[t]{\colw}{#3}\par + \bigskip +} + +%% display the bibliography +\long\def\BBlography{ + \newlength{\colw} + \setlength{\colw}{\textwidth} + \addtolength{\colw}{-\rw} + \IfFileExists{bibliography.BBlog.tex}{ + \input{bibliography.BBlog}}{{\tt error: missing BBlog bibliography file}} +} diff --git a/Figs/3x2_skel.pdf b/Figs/3x2_skel.pdf new file mode 100644 index 0000000..bb4c7cd Binary files /dev/null and b/Figs/3x2_skel.pdf differ diff --git a/Figs/3x2_subskel.pdf b/Figs/3x2_subskel.pdf new file mode 100644 index 0000000..153c205 Binary files /dev/null and b/Figs/3x2_subskel.pdf differ diff --git a/Figs/Lshape.pdf b/Figs/Lshape.pdf new file mode 100644 index 0000000..a482514 Binary files /dev/null and b/Figs/Lshape.pdf differ diff --git a/Figs/Lshape_label.pdf b/Figs/Lshape_label.pdf new file mode 100644 index 0000000..fe692bf Binary files /dev/null and b/Figs/Lshape_label.pdf differ diff --git a/Figs/bethe.pdf b/Figs/bethe.pdf new file mode 100644 index 0000000..7a02d88 Binary files /dev/null and b/Figs/bethe.pdf differ diff --git a/Figs/boundary_circuit.pdf b/Figs/boundary_circuit.pdf new file mode 100644 index 0000000..cbe4ac8 Binary files /dev/null and b/Figs/boundary_circuit.pdf differ diff --git a/Figs/boundary_circuit_label.pdf b/Figs/boundary_circuit_label.pdf new file mode 100644 index 0000000..39132a2 Binary files /dev/null and b/Figs/boundary_circuit_label.pdf differ diff --git a/Figs/boundary_circuit_naked.pdf b/Figs/boundary_circuit_naked.pdf new file mode 100644 index 0000000..9e8b7fa Binary files /dev/null and b/Figs/boundary_circuit_naked.pdf differ diff --git a/Figs/circuit.pdf b/Figs/circuit.pdf new file mode 100644 index 0000000..c7c3d44 Binary files /dev/null and b/Figs/circuit.pdf differ diff --git a/Figs/comb.pdf b/Figs/comb.pdf new file mode 100644 index 0000000..b3d3d6c Binary files /dev/null and b/Figs/comb.pdf differ diff --git a/Figs/enclosed_example.pdf b/Figs/enclosed_example.pdf new file mode 100644 index 0000000..2df6492 Binary files /dev/null and b/Figs/enclosed_example.pdf differ diff --git a/Figs/enclosed_example_auxiliary_cover.pdf b/Figs/enclosed_example_auxiliary_cover.pdf new file mode 100644 index 0000000..edfd9d8 Binary files /dev/null and b/Figs/enclosed_example_auxiliary_cover.pdf differ diff --git a/Figs/enclosed_example_cover.pdf b/Figs/enclosed_example_cover.pdf new file mode 100644 index 0000000..a26c5a6 Binary files /dev/null and b/Figs/enclosed_example_cover.pdf differ diff --git a/Figs/enclosed_example_directed.pdf b/Figs/enclosed_example_directed.pdf new file mode 100644 index 0000000..4244482 Binary files /dev/null and b/Figs/enclosed_example_directed.pdf differ diff --git a/Figs/enclosed_example_monomer_auxiliary_cover.pdf b/Figs/enclosed_example_monomer_auxiliary_cover.pdf new file mode 100644 index 0000000..19a91cf Binary files /dev/null and b/Figs/enclosed_example_monomer_auxiliary_cover.pdf differ diff --git a/Figs/enclosed_example_monomer_cover.pdf b/Figs/enclosed_example_monomer_cover.pdf new file mode 100644 index 0000000..80e6f38 Binary files /dev/null and b/Figs/enclosed_example_monomer_cover.pdf differ diff --git a/Figs/etareplace_cross.pdf b/Figs/etareplace_cross.pdf new file mode 100644 index 0000000..fa9d840 Binary files /dev/null and b/Figs/etareplace_cross.pdf differ diff --git a/Figs/etareplace_empty.pdf b/Figs/etareplace_empty.pdf new file mode 100644 index 0000000..48f6deb Binary files /dev/null and b/Figs/etareplace_empty.pdf differ diff --git a/Figs/etareplace_occupied.pdf b/Figs/etareplace_occupied.pdf new file mode 100644 index 0000000..ef11b35 Binary files /dev/null and b/Figs/etareplace_occupied.pdf differ diff --git a/Figs/etareplace_r.pdf b/Figs/etareplace_r.pdf new file mode 100644 index 0000000..70d7dba Binary files /dev/null and b/Figs/etareplace_r.pdf differ diff --git a/Figs/even_example.pdf b/Figs/even_example.pdf new file mode 100644 index 0000000..6751c57 Binary files /dev/null and b/Figs/even_example.pdf differ diff --git a/Figs/even_example_auxcover.pdf b/Figs/even_example_auxcover.pdf new file mode 100644 index 0000000..e7a7e37 Binary files /dev/null and b/Figs/even_example_auxcover.pdf differ diff --git a/Figs/even_example_auxiliary.pdf b/Figs/even_example_auxiliary.pdf new file mode 100644 index 0000000..cce5f35 Binary files /dev/null and b/Figs/even_example_auxiliary.pdf differ diff --git a/Figs/even_example_cover.pdf b/Figs/even_example_cover.pdf new file mode 100644 index 0000000..2985cd9 Binary files /dev/null and b/Figs/even_example_cover.pdf differ diff --git a/Figs/even_example_label.pdf b/Figs/even_example_label.pdf new file mode 100644 index 0000000..3c32dc9 Binary files /dev/null and b/Figs/even_example_label.pdf differ diff --git a/Figs/even_example_label_pre.pdf b/Figs/even_example_label_pre.pdf new file mode 100644 index 0000000..5ba8b2c Binary files /dev/null and b/Figs/even_example_label_pre.pdf differ diff --git a/Figs/even_example_monomer.pdf b/Figs/even_example_monomer.pdf new file mode 100644 index 0000000..cae96f9 Binary files /dev/null and b/Figs/even_example_monomer.pdf differ diff --git a/Figs/extended_dimer.pdf b/Figs/extended_dimer.pdf new file mode 100644 index 0000000..8fc0b67 Binary files /dev/null and b/Figs/extended_dimer.pdf differ diff --git a/Figs/extended_dimer_triangle.pdf b/Figs/extended_dimer_triangle.pdf new file mode 100644 index 0000000..efc0885 Binary files /dev/null and b/Figs/extended_dimer_triangle.pdf differ diff --git a/Figs/minimal_circuit.pdf b/Figs/minimal_circuit.pdf new file mode 100644 index 0000000..06fa896 Binary files /dev/null and b/Figs/minimal_circuit.pdf differ diff --git a/Figs/noncoverable.pdf b/Figs/noncoverable.pdf new file mode 100644 index 0000000..754d434 Binary files /dev/null and b/Figs/noncoverable.pdf differ diff --git a/Figs/oddrepl_aft.pdf b/Figs/oddrepl_aft.pdf new file mode 100644 index 0000000..f4924a0 Binary files /dev/null and b/Figs/oddrepl_aft.pdf differ diff --git a/Figs/oddrepl_bef.pdf b/Figs/oddrepl_bef.pdf new file mode 100644 index 0000000..3eb7995 Binary files /dev/null and b/Figs/oddrepl_bef.pdf differ diff --git a/Figs/simple_example.pdf b/Figs/simple_example.pdf new file mode 100644 index 0000000..3aca93a Binary files /dev/null and b/Figs/simple_example.pdf differ diff --git a/Figs/simple_example_auxiliary.pdf b/Figs/simple_example_auxiliary.pdf new file mode 100644 index 0000000..9282f37 Binary files /dev/null and b/Figs/simple_example_auxiliary.pdf differ diff --git a/Figs/simple_example_labels.pdf b/Figs/simple_example_labels.pdf new file mode 100644 index 0000000..dd7ce8c Binary files /dev/null and b/Figs/simple_example_labels.pdf differ diff --git a/Figs/simple_example_monomers.pdf b/Figs/simple_example_monomers.pdf new file mode 100644 index 0000000..74738db Binary files /dev/null and b/Figs/simple_example_monomers.pdf differ diff --git a/Figs/triangle_3.pdf b/Figs/triangle_3.pdf new file mode 100644 index 0000000..696649c Binary files /dev/null and b/Figs/triangle_3.pdf differ diff --git a/Figs/triangle_3_Xcover1.pdf b/Figs/triangle_3_Xcover1.pdf new file mode 100644 index 0000000..7516045 Binary files /dev/null and b/Figs/triangle_3_Xcover1.pdf differ diff --git a/Figs/triangle_3_Xcover3.pdf b/Figs/triangle_3_Xcover3.pdf new file mode 100644 index 0000000..8f07a93 Binary files /dev/null and b/Figs/triangle_3_Xcover3.pdf differ diff --git a/Figs/triangle_3_cover1.pdf b/Figs/triangle_3_cover1.pdf new file mode 100644 index 0000000..c200454 Binary files /dev/null and b/Figs/triangle_3_cover1.pdf differ diff --git a/Figs/triangle_3_cover3.pdf b/Figs/triangle_3_cover3.pdf new file mode 100644 index 0000000..05bcc4c Binary files /dev/null and b/Figs/triangle_3_cover3.pdf differ diff --git a/Figs/triangle_3_pre.pdf b/Figs/triangle_3_pre.pdf new file mode 100644 index 0000000..fe121fd Binary files /dev/null and b/Figs/triangle_3_pre.pdf differ diff --git a/Figs/triangle_4.pdf b/Figs/triangle_4.pdf new file mode 100644 index 0000000..d779fa3 Binary files /dev/null and b/Figs/triangle_4.pdf differ diff --git a/Figs/triangle_4_pre.pdf b/Figs/triangle_4_pre.pdf new file mode 100644 index 0000000..c38fd0c Binary files /dev/null and b/Figs/triangle_4_pre.pdf differ diff --git a/Figs/triangle_5.pdf b/Figs/triangle_5.pdf new file mode 100644 index 0000000..3f0bc4e Binary files /dev/null and b/Figs/triangle_5.pdf differ diff --git a/Figs/triangle_5_directed.pdf b/Figs/triangle_5_directed.pdf new file mode 100644 index 0000000..f98fb40 Binary files /dev/null and b/Figs/triangle_5_directed.pdf differ diff --git a/Figs/triangle_5_pre.pdf b/Figs/triangle_5_pre.pdf new file mode 100644 index 0000000..3a911ff Binary files /dev/null and b/Figs/triangle_5_pre.pdf differ diff --git a/Figs/triangle_groups_l.pdf b/Figs/triangle_groups_l.pdf new file mode 100644 index 0000000..86aa908 Binary files /dev/null and b/Figs/triangle_groups_l.pdf differ diff --git a/Figs/triangle_groups_lr.pdf b/Figs/triangle_groups_lr.pdf new file mode 100644 index 0000000..cbe3ddf Binary files /dev/null and b/Figs/triangle_groups_lr.pdf differ diff --git a/Figs/triangle_groups_r.pdf b/Figs/triangle_groups_r.pdf new file mode 100644 index 0000000..772012c Binary files /dev/null and b/Figs/triangle_groups_r.pdf differ diff --git a/Giuliani_Jauslin_Lieb_2015.tex b/Giuliani_Jauslin_Lieb_2015.tex new file mode 100644 index 0000000..d383c37 --- /dev/null +++ b/Giuliani_Jauslin_Lieb_2015.tex @@ -0,0 +1,1642 @@ +\documentclass{kiss} +% load packages +\usepackage{header} +% bibliography commands +\usepackage{BBlog} +% miscellaneous commands +\usepackage{toolbox} +% main style file +\usepackage{iansecs} + +\definecolor{darkgreen}{HTML}{32CD32} + +\def\defd#1{{\bf #1}} + +\begin{document} + +\pagestyle{empty} +{\LARGE\bf\hfil A Pfaffian formula\par +\bigskip +\hfil for monomer-dimer partition functions}\par +\hugeskip + +{\bf\hfil Alessandro Giuliani, Ian Jauslin, Elliott H.~Lieb}\par +\hugeskip +\hfil 2015 +\hugeskip + +\indent We consider the monomer-dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the vertices and edges of the graph. + +\hugeskip + +\tableofcontents + +\vfill +\eject + +\pagestyle{plain} +\setcounter{page}1 + +\section{Introduction} + +\indent The monomer-dimer problem is one of the important classical structures in statistical mechanics and computer science. It starts with a {\it graph} $g$, which is a collection of points, called {\it vertices}, and lines, called {\it edges}, between specified pairs of points. A dimer is an object that occupies a single edge and its endpoints, and a monomer is an object that occupies a single vertex. A {monomer-dimer covering} of $g$ (hereafter refered to as an MD covering) is a collection of monomers and dimers (that is to say vertices and edges) such that every vertex is covered by exactly one of these objects, that is by either a monomer or a dimer. Note that, in any MD covering, the number of vertices in $g$ is equal to the number of monomers plus twice the number of dimers. The present work is devoted to {\it planar} graphs (which are those that can be drawn in $\mathbb R^2$ without edge crossings). + +\indent The classical problem associated with MD overings is their enumeration at fixed number of monomers. The object of this work is to find formulas for the generating function of this enumeration problem: +\begin{equation} +\Xi(z):=\sum_{\mathrm{MD\ coverings}}z^{\mathrm{number\ of\ monomers}}. +\label{eqXizdef}\end{equation} +Clearly, $\Xi(z)$ is a polynomial in $z$, and $z$ is called the {\it monomer fugacity}. This polynomial has all its zeros on the imaginary axis~[\cite{HL70}, \cite{HL72}]. In addition, the summation in~(\ref{eqXizdef}) can be generalized by assigning weights to edges and/or vertices. + +\indent In the pure dimer case, where $z=0$, $\Xi$ has been shown by Temperley and Fisher (for the square lattice) [\cite{TF61}] and by Kasteleyn (for general planar graphs) [\cite{Ka63}] to be expressible as a Pfaffian (which is convenient since Pfaffians can be computed as square roots of determinants). However, when monomers are allowed to appear, such a Pfaffian formula is thought to be impossible (at least a Pfaffian formula for the {\it full} MD problem on {\it any} planar graphs): indeed it has been shown [\cite{Je87}] that the enumeration of MD coverings of generic planar graphs is ``computationally intractable'', whereas Pfaffians can be computed in polynomial time. More precisely, [\cite{Je87}] proves that the enumeration of MD coverings of generic planar graphs is +``$\# P$ complete'', which implies that it is believed not to be computable in polynomial time. + +\indent However, by introducing restrictions on the location of monomers, such a result can be proven in some cases. Namely, in [\cite{TW03}, \cite{Wu06}], the authors use a Pfaffian formula to compute the partition function of a system with a {\it single} monomer located on the boundary of a finite square lattice. In [\cite{PR08}], the MD problem is studied on the square lattice on the half-plane with the restriction that the monomers are {\it fixed} on points of the boundary. They derive a Pfaffian formula for this case, and use it to compute the scaling limit of the multipoint boundary monomer correlations. Finally, in [\cite{AF14}], it is shown that if the monomers are {\it fixed} at any position in a square lattice, then the partition function can also be written as a Pfaffian. In addition, the authors use their formula for +computing the asymptotics of monomer correlations, thus generalizing the classical results by Fisher, Hartwig and Stephenson [\cite{FS63}, \cite{Ha66}, \cite{FH69}] +for monomer correlations along a row, column or main diagonal. + +\indent In the present work, we prove that, on an {\it arbitrary} planar graph, the {\it boundary} MD partition function (in which the monomers are restricted to the boundary of the graph, but are not necessarily fixed at prescribed locations) with arbitrary dimer and monomer weights can be written as a Pfaffian. +By differentiation with respect to the monomer weights, we also obtain a fermionic Wick rule for boundary monomer correlations. +There are two main novelties in our result. First, we obtain the Pfaffian formula for the partition function with arbitrary weights rather than for +the problem with monomers at fixed locations. Second, we study the boundary MD problem on {\it any} planar graph, +in particular for graphs that may not admit a transfer matrix treatment. +\bigskip + +{\bf Remarks}: +\listparpenalty +\begin{itemize} +\item An alternative approach for the boundary MD problem on an $N\times M$ rectangle (by which we mean $N$ vertices times $M$ vertices) with monomers allowed only on the upper and lower sides +would be to use the transfer matrix technique [\cite{Li67}]. In that case, the boundary MD partition function is written as $x_M\cdot V^{M-1}x_1$ where $V$ is the +($N\times N$) transfer matrix and $x_1$ and $x_M$ are vectors determined by the boundary condition at the boundaries $y=1$ and $y=M$ respectively. +Since the monomers are only allowed on these two boundaries, the matrix $V$ is the transfer matrix for {\it pure} dimer coverings, which can be +diagonalized as in [\cite{Li67}]. The partition function can then be computed by setting the vectors $x_1$ and $x_M$ appropriately. +\item The boundary monomer correlations at close-packing are critical, in that if the graph is ``regular enough'' (e.g., +if it is a finite portion of a lattice) they decay polynomially at large distances, +like $1/(distance)$, asymptotically as the size of the graph tends to infinity. +See [\cite{PR08}] for a proof of this fact on the square lattice on the half-plane. +If the graph is a discrete, regular, approximation of a finite domain of $\mathbb R^2$, +the scaling limit of these correlations +is expected to exist and to be conformally invariant under conformal mappings of the domain, in analogy with other observables +of the critical 2D Ising model and of the close-packed dimer model [\cite{Ke00}, \cite{Ke01}, \cite{Sm01}, \cite{Sm10}, \cite{CHI15}, \cite{Du11}, \cite{Du15}]. +In particular, they are expected to coincide with those of complex chiral free fermions [\cite{PR08}]. +It is unclear whether this scaling limit is stable under perturbations violating planarity (e.g., under the addition of small dimer weights along extra +edges crossings). Our Pfaffian formula offers a starting point for a perturbative multiscale analysis of the problem, in the spirit of [\cite{PS}, \cite{Sp00}, \cite{GGM12}, \cite{GMT15b}, \cite{GMT15}]. +\item An alternative approach for the boundary MD problem on generic +planar graphs is via the random current representation developed by +Aizenman [\cite{Ai82}]. It has been recently observed +[\cite{AD}] that this representation, adapted to planar lattices, implies, for +purely geometrical +reasons, the validity of the fermionic Wick rule both for boundary spin +correlations in the nearest neighbor Ising model, and might imply the +same for boundary monomer correlations in the dimer model. +Their method also suggests a stochastic geometric perspective on the +emergence of planarity at the critical points of non-planar 2D models, in the sense of the previous item. +\item It may be worth noting that the MD partition function can be computed exactly in some cases, e.g. on the complete graph [\cite{ACM14}]. +\end{itemize} +\unlistparpenalty + +\indent We will now state our main result more precisely, for which we need some notation. Let $\mathcal G$ denote the set of finite planar graphs with edge-weights and vertex-weights, embedded in $\mathbb R^2$, that have an even number of vertices, and contain no {\it double edges} or {\it self-contractions} (that is the endpoints of an edge are distinct, and no two edges share the same endpoints). Note that these graphs are not necessarily connected. The +evenness condition is not restrictive, in that a graph with an odd number of vertices can always be reduced to an even one, by adding an isolated (disconnected) vertex. + +\indent Given $g\in\mathcal G$, its \defd{boundary graph} $\partial g$ is defined as the sub-graph of $g$ containing the edges and vertices that can be connected to infinity without crossing any edge of $g$ (here we say that an edge can be connected to infinity without crossing any other edge, if a point at the center of the edge can be). The set of vertices of $g$ is denoted by $\mathcal V(g)$ and its set of edges by $\mathcal E(g)$. The edge linking two vertices $v_1$ and $v_2$ will be denoted by $\{v_1,v_2\}\equiv\{v_2,v_1\}$. The weight of a vertex $v\in\mathcal V(g)$ (i.e. the fugacity of a monomer located at $v$) is denoted by $\ell_v$ and the weight of an edge $\{v_1,v_2\}\in\mathcal E(g)$ (i.e. the fugacity of a dimer located at $\{v_1,v_2\}$) is denoted by $d_{\{v_1,v_2\}}\equiv d_{v_1,v_2}$. The number of vertices in $g$ in denoted by $|g|$. +In the following we will often consider {\it directed} graphs, which are obtained by assigning a direction to every edge: if the edge $\{v_1,v_2\}$ is directed from $v_1$ to $v_2$, we write $v_1\succ v_2$. + +\indent The set of MD coverings of $g$ is denoted by $\Omega(g)$ and the set of pure dimer coverings by $\Omega_0(g)$. Given an MD covering $\sigma\in\Omega(g)$, we denote the set of vertices covered by monomers by $\mathfrak M(\sigma)\subset\mathcal V(g)$ and the set of edges covered by dimers by $\mathfrak D(\sigma)\subset\mathcal E(g)$. The \defd{boundary MD partition function} of a graph $g$ is defined as the partition function of MD coverings of $g$ in which the monomers are restricted to vertices of $\partial g$: +\begin{equation} +\Xi_\partial(\bm\ell,\mathbf d):=\sum_{\displaystyle\mathop{\scriptstyle\sigma\in\Omega(g)}_{\mathfrak M(\sigma)\subset\mathcal V(\partial g)}}\prod_{v\in\mathfrak M(\sigma)}\ell_v\prod_{e\in\mathfrak D(\sigma)}d_e. +\label{eqpartfnboundary}\end{equation} +Note that restricting the monomers of $\sigma$ to be on boundary vertices can be enforced by setting all other $\ell_v$'s to 0. + +\indent The Pfaffian of a $2n\times2n$-dimensional antisymmetric matrix $(A_{i,j})$ +is defined as +\begin{equation} + \mathrm{pf}(A):=\frac1{2^nn!}\sum_{\pi\in\mathcal +S_{2n}}(-1)^\pi\prod_{i=1}^nA_{\pi(2i-1),\pi(2i)} +\label{eqpfaffdef}\end{equation} +where $\mathcal S_{2n}$ denotes the set of permutations of $\{1,\cdots,2n\}$ and $(-1)^\pi$ is the signature of $\pi\in\mathcal S_{2n}$. +\bigskip + +\theoname{Theorem}{Main result}\label{theomain} +For every $g\in\mathcal G$, the edges of $g$ can be directed and its vertices labeled $(v_1,\cdots,v_{|g|})$ in such a way that, by defining +\begin{equation} +a_{i,j}(\mathbf d):=\left\{ +\begin{array}{l@{\ }l} + +d_{v_i,v_j}&\mathrm{if\ }\{v_i,v_j\}\in\mathcal E(g)\ \mathrm{and\ }v_i\succ v_j\\ + -d_{v_i,v_j}&\mathrm{if\ }\{v_i,v_j\}\in\mathcal E(g)\ \mathrm{and\ }v_i\prec v_j\\ + 0&\mathrm{otherwise} +\end{array} +\right. +\label{eqAdef}\end{equation} +and, for $ij$, we have +\begin{equation} +\mathrm{pf}(A(\bm \ell))=\sum_{k=0}^{N/2}\sum_{\displaystyle\mathop{ +\scriptstyle\mathcal +I\subset\{1,\cdots,N\}}_{|\mathcal +I|=2k}}\mathrm{pf}(\left[a\right]_{\mathcal I})\prod_{i\in\mathcal +I}\ell_i +\label{eqliebtheo}\end{equation} +in which +$\left[a\right]_{\mathcal I}$ denotes the matrix obtained from $a$ by +removing the $i$-th line and column for every $i$ in $\mathcal I$, and if $\mathcal I=\{1,\cdots,N\}$, then $\mathrm{pf}([a]_{\{1,\cdots,N\}})\equiv1$. +\endtheo +\medskip + +\indent In [\cite{Li68}], the theorem was proved in the case in which the $\ell_i$ +are equal, but the proof is immediately generalizable to arbitrary +$\bm \ell$. The only change needed in the proof of [\cite{Li68}] is to change +equation [\cite{Li68}, (21)] from +\begin{equation} +\frac1Z\mathrm{trace}\left(\prod_{i=1}^N(\lambda+C_i)\right) +\qquad\mathrm{to}\qquad +\frac1Z\mathrm{trace}\left(\prod_{i=1}^N(\ell_i+C_i)\right). +\label{eqreplLi}\end{equation} +The rest of the proof is identical. +\bigskip + +\indent If we let all the $\ell_i$'s +equal $z$, then $\mathrm{pf}(A(\bm \ell))$ in~(\ref{eqliebtheo}) is the +polynomial in $z$ whose $2k$-coefficient is the sum of all sub-Pfaffians of +order $2k$. If $a_{i,j}$ is defined as in~(\ref{eqadef}), this {\it seems} +to count all MD coverings with $2k$ monomers: indeed, by Kasteleyn's +theorem, $\mathrm{pf}([a]_{\mathcal I})$ {\it appears} to be the partition +function of dimer coverings of the graph $[g]_{\mathcal I}$ obtained from +$g$ by removing the vertices whose labels are in $\mathcal I$, or +equivalently of MD coverings with monomers on the vertices whose labels are +in $\mathcal I$. This is not the case, however, {\it since $[g]_{\mathcal I}$ is +not necessarily a positive Kasteleyn graph}. + +\indent In the rest of this paper, we will provide an algorithm to direct and label +$g$ in such a way that when the vertices in $\mathcal I$ are restricted to +the boundary, which is imposed by setting all other $\ell_i$'s to zero, +$[g]_{\mathcal I}$ is a positive Kasteleyn graph. In that case, +$\mathrm{pf}(A(\bm\ell,\mathbf d))$ is the boundary MD partition function +with $A(\bm\ell,\mathbf d)$ defined in~(\ref{eqpolysta}). +\bigskip + +\point{\bf Lower bound on the monomer-dimer partition function.} +When $\mathrm{pf}(A(\bm\ell,\mathbf d))$ does {\it not} equal the MD partition +function, it is so either because the terms in a sub-Pfaffian +$[a(\mathbf d)]_{\mathcal I}$ do not add up constructively, or because the sign of +$\mathrm{pf}([a(\mathbf d)]_{\mathcal I})$ is wrong. Nevertheless, the following +theorem holds. +\bigskip + +\theoname{Theorem}{Lower bound for the terms in the MD partition function}\label{theolowerbound} +For every $g\in\mathcal G$, if $d_e\ge0$ for all $e\in\mathcal +E(g)$, then defining +\begin{equation} + a_{i,j}(\mathbf d):=\left\{ + \begin{array}{l@{\ }l} + +d_{v_i,v_j}&\mathrm{if\ }\{v_i,v_j\}\in\mathcal E(g)\ \mathrm{and\ }v_i\succ v_j\\ + -d_{v_i,v_j}&\mathrm{if\ }\{v_i,v_j\}\in\mathcal E(g)\ \mathrm{and\ }v_i\prec v_j\\ + 0&\mathrm{otherwise} + \end{array} + \right. +\label{eqAsimpledef}\end{equation} +in which $v_i$ is a shorthand for $\omega^{-1}(i)$ and, for $i{\displaystyle}l} +M_n(i_1,\cdots,i_{2n})=& +(-1)^n\mathrm{pf}(\{a^{-1}(\mathbf d)\}_{\mathcal I})\\[0.3cm] +=&\frac{1}{2^nn!}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{j=1}^n(-a^{-1}(\mathbf d))_{i_{\pi(2j-1)},i_{\pi(2j)}}, +\end{array}\label{eqpfMconc}\end{equation} +which concludes the proof, by noting that $(-a^{-1}(\mathbf d))_{i,j}=M_1(i,j)$. +\hfill$\square$ + + + + + +\section{Simple graphs}\label{secsimple} + +\indent We shall first consider a family of graphs $\mathcal G_s\subset\mathcal G$, +called \defd{simple graphs}, consisting of a simple, closed circuit of +vertices, which may or may not be connected to each other through edges +inside the circuit (see figure~\ref{figsimpleexample} for an example). All +vertices are therefore part of the circuit, which means that we +are computing the full MD partition function on the graph (with no +restriction on the monomer locations). We denote the +sub-graph of $g$ obtained by removing all internal lines by $\partial g$. +\bigskip + +\begin{figure} +\hfil\parbox[m]{3cm}{\includegraphics[width=3cm]{Figs/simple_example.pdf}} +\hfil\parbox[m]{4cm}{\includegraphics[width=4cm]{Figs/bethe.pdf}}\par\penalty10000 +\bigskip +\captionshort{Examples of simple graphs. One is a {\it Bethe lattice} built on a +square lattice.} +\label{figsimpleexample}\end{figure} + +\indent One way to think about a simple graph is as a special kind of Hamiltonian +graph, i.e., a graph with a closed circuit that visits every vertex exactly +once. Such a graph can be represented as a circuit of +vertices, $\partial g$, +with connections both external and internal of these vertices. These are +the only vertices, as in our simple graphs, but we impose the additional +constraint on the Hamiltonian graph that all other edges are exclusively +inside or exclusively outside $\partial {g}$. See figure~\ref{fighamiltonianexample} for an example. +\bigskip + +\begin{figure} +\hfil\includegraphics[width=5cm]{Figs/comb.pdf}\par\penalty10000 +\medskip +\caption{This simple graph can be seen as a sub-graph of the $10\times7$ rectangle, in which we discarded the edges that are on the outside of the grey Hamilton cycle.} +\label{fighamiltonianexample} +\end{figure} + +\indent We will now describe how to direct and label a simple graph in such a way that its MD partition function can be written as a Pfaffian as in theorem~\ref{theomain}. +\bigskip + +\delimtitleref{Directing and labeling a simple graph}\label{recipesimple} +We label the vertices of $g$ following the edges of $\partial g$ +sequentially in the counterclockwise direction +(see figure~\ref{figsimpleexamplelabels} for an example). The resulting +labeling is denoted by $\omega$. The location of the vertex labeled as 1 is +unimportant. The graph is directed by setting $v\succ v'$ if and +only if $\omega(v)<\omega(v')$. +\enddelim +\bigskip + +\begin{figure} +\hfil\includegraphics[width=4cm]{Figs/simple_example_labels.pdf}\par\penalty10000 +\medskip +\captionshort{The labeling and direction for the first simple graph in +figure~\ref{figsimpleexample}.} +\label{figsimpleexamplelabels}\end{figure} +\bigskip + +\indent One readily checks that the resulting +directed graph is Kasteleyn: indeed each minimal circuit, each having +an empty interior, is labeled in an increasing fashion and is thus {\it good}. +In addition, by proposition~\ref{propkasteleyn}, it is positive since +$$ +\sigma=\{\big(\omega^{-1}(1),\omega^{-1}(2)\big),\cdots,\big(\omega^{-1}(|g|-1), +\omega^ { -1}(|g|)\big) \} +$$ +is a dimer covering of $g$, and its associated permutation is +$\pi_\sigma^{(\omega)}=\mathds1$ (see definition~\ref{defpositivity}). +\bigskip + +\indent If the monomers of an MD covering are fixed, the possible dimer +positions are the possible (pure) dimer coverings of the sub-graph of $g$ +obtained by removing the vertices that have a monomer. We will now prove +that such a sub-graph is Kasteleyn and positive which implies that the dimer +partition function on it satisfies the Pfaffian +formula~(\ref{eqkasteleyntheo}) which can be substituted +in~(\ref{eqliebtheo}) to reproduce the MD partition function. + +\indent Given a family +of monomers $\mathcal M\subsetneq\mathcal V(g)$ of even cardinality, we +define $[g]_{\mathcal M}$ as the sub-graph of $g$ obtained by removing the +vertices in $\mathcal M$ (which may be disconnected), direct the edges of $[g]_{\mathcal M}$ in the same way as those of $g$ (noticing that $\mathcal E([g]_{\mathcal M})\subset\mathcal E(g)$), and define a new labeling $\omega_{\mathcal M}$ +for its vertices that is obtained from $\omega$ by removing the monomeric +vertices and eliminating the labeling gaps: +\begin{equation} +\omega_{\mathcal M}(v):=\omega(v)-\mathrm{card}\{v'\in\mathcal M\quad|\quad +\omega(v')<\omega(v)\}. +\label{eqinducedlabeling}\end{equation} +See figure~\ref{figsimpleexamplemonomers} for an +example. +\bigskip + +\begin{figure} +\hfil\includegraphics[width=4cm]{Figs/simple_example_monomers.pdf} +\hfil\includegraphics[width=4cm]{Figs/simple_example_auxiliary.pdf}\par\penalty10000 +\medskip +\caption{The sub-graph $[g]_{\mathcal M}$ and the auxiliary graph $\gamma_{\mathcal M}$ for the graph in +figure~\ref{figsimpleexamplelabels} with +monomers on the vertices labeled by 1 and 5.} +\label{figsimpleexamplemonomers}\end{figure} +\bigskip + +\theo{Lemma}\label{lemmasimplepositivity} +For every $g\in\mathcal G_s$, directed and labeled as above, for all +$\mathcal M\subsetneq\mathcal V(g)$ of even cardinality $|\mathcal M|$, the +sub-graph +$[g]_{\mathcal M}$ whose vertices are labeled by $\omega_{\mathcal M}$ is +Kasteleyn and positive. +\endtheo +\medskip + +\indent\underline{Proof}: We prove the lemma by constructing an auxiliary graph +$\gamma_{\mathcal M}$, which is a super-graph of $[g]_{\mathcal M}$ (i.e. +$[g]_{\mathcal M}$ is a sub-graph of $\gamma_{\mathcal M}$), by adding extra +edges in such a way that $\gamma_{\mathcal M}$ is Kasteleyn, and that its +positivity is obvious. The positivity of +$[g]_{\mathcal M}$ then follows from proposition~\ref{propkasteleyn}. + +{\bf Remark}: By adding edges to the graph $[g]_{\mathcal M}$ and/or +changing weights, we change the partition function. This, however, is of no +consequence since the extra edges are only used to prove the positivity of +the dimer coverings of $\gamma_{\mathcal M}$ that are {\it also} +coverings of $[g]_{\mathcal M}$ (that is, coverings that do not +use the extra edges), which are the ones of actual interest. + +\indent The auxiliary graph $\gamma_{\mathcal M}$ is obtained from $[g]_{\mathcal +M}$ by +\begin{itemize} +\item adding the edge $\{\omega_{\mathcal M}^{-1}(i),\omega_{\mathcal +M}^{-1}(i+1)\}$ for $1\le i\le|[g]_{\mathcal M}|-1$ whenever it is +not present in $[g]_{\mathcal M}$, and directing it from $\omega_{\mathcal +M}^{-1}(i)$ to $\omega_{\mathcal M}^{-1}(i+1)$, +\item adding the edge $\{\omega_{\mathcal M}^{-1}(1),\omega_{\mathcal +M}^{-1}(|[g]_{\mathcal M}|)\}$ if it is not present in $[g]_{\mathcal M}$, +and directing it from $\omega_{\mathcal M}^{-1}(1)$ to $\omega_{\mathcal +M}^{-1}(|[g]_{\mathcal M}|)$. +\end{itemize} +See figure~\ref{figsimpleexamplemonomers} for an example. + +\indent This graph is Kasteleyn and positive for the same reason $g$ is (see above). +By definition, this implies that every dimer covering of $\gamma_{\mathcal M}$ is positive, and since $\gamma_{\mathcal M}$ is a super-graph of +$[g]_{\mathcal M}$ and has the same vertices (which are labeled in the same +way), every dimer covering of $[g]_{\mathcal M}$ is a dimer covering of +$\gamma_{\mathcal M}$. This concludes the proof of the lemma.\hfill$\square$ +\bigskip + +\indent It follows from lemma~\ref{lemmasimplepositivity} and +theorem~\ref{theokasteleyn} that, for $\mathcal M\neq\mathcal V(g)$, by defining $a(\mathbf d)$ as in~(\ref{eqAdef}), +$\mathrm{pf}([a(\mathbf d)]_{\mathcal M})$ is the partition function of dimers on $[g]_{\mathcal M}$. +By theorem~\ref{theolieb}, this implies theorem \ref{theomain} for +simple graphs, provided $g\in\mathcal G_s$ is directed and labeled as above (see ``\ref{recipesimple}''). + +\section{Additional preliminaries}\label{secprelin} +In this section, we discuss several properties which we will use to treat graphs that have vertices inside their boundary. + + +\subsection{Properties of Kasteleyn graphs}\label{subsecpropkasteleyn} +In this section, we prove a few useful lemmas about Kasteleyn graphs. The key result is the following. +\bigskip + +\theo{Lemma}\label{lemmacombcircuits} +Given two circuits $c_1$ and $c_2$ as in~(\ref{eqmergecircuit}), if $c_1$ and $c_2$ are good then their merger $c=c_1\Delta c_2$ is as well. +\endtheo +\medskip + +\indent\underline{Proof}: We write $c_1$ and $c_2$ as in~(\ref{eqmergecircuit}): +$$ +c_1=(v_1,\cdots,v_{\nu+1},v_{\nu+2},\cdots,v_{|c_1|}),\quad +c_2=(v_{\nu+1},\cdots,v_{1},v_{\nu+2}',\cdots,v_{|c_2|}') +$$ +and +$$ +c=(v_{\nu+1},\cdots,v_{|c_1|},v_1,v_{\nu+2}',\cdots,v_{|c_2|}'). +$$ + +\indent We first prove that +\begin{itemize} +\item if $\nu$ is odd, then + \begin{itemize} + \item if $c_1$ and $c_2$ are either both oddly-directed or both evenly-directed then $c$ is +oddly-directed, + \item otherwise $c$ is evenly-directed. + \end{itemize} +\item if $\nu$ is even, then + \begin{itemize} + \item if $c_1$ and $c_2$ are either both oddly-directed or both evenly-directed then $c$ is evenly-directed, + \item otherwise $c$ is oddly-directed. + \end{itemize} +\end{itemize} + +\indent Indeed, a circuit $c_1$ is oddly-directed if and only if the numbers of backwards edges in +$(v_1,\cdots,v_{\nu+1})$ and in $(v_{\nu+1},\cdots,v_{|c_1|},v_1)$ have +different parity (by which we mean evenness or oddness), and +$c_2$ is oddly-directed if and only if the numbers of backwards edges in +$(v_{\nu+1},\cdots,v_{1})$ and in +$(v_1,v_{\nu+2}',\cdots,v_{|c_2|}',v_{\nu+1})$ have different parity. +Therefore, if $\nu$ is even, using the fact that the numbers of backwards +edges in $(v_1,\cdots,v_{\nu+1})$ and in $(v_{\nu+1},\cdots,v_1)$ have the +same parity, it follows that $c_1\Delta c_2$ is oddly-directed if and only if $c_1$ is +oddly-directed and $c_2$ is evenly-directed or vice-versa. If $\nu$ is odd, the numbers of +backwards edges in $(v_1,\cdots,v_{\nu+1})$ and in $(v_{\nu+1},\cdots,v_1)$ +have different parity, so that $c_1\Delta c_2$ is oddly-directed if and only if $c_1$ +and $c_2$ are either both oddly-directed or both evenly-directed. +\medskip + +\indent The proof of the lemma is then concluded by noticing that +if $\nu$ is odd then the number of vertices that are encircled by either +$c_1$ or $c_2$ has the same parity as the number of vertices encircled by +$c$, whereas the parity is different if $\nu$ is even.\hfill$\square$ +\bigskip + +\indent Lemma \ref{lemmacombcircuits} has a number of useful consequences, which are discussed in the following. +\medskip + +{\bf Proof of lemma~\ref{lemmakasteleyncomplete}}: Given a circuit $c$ of a Kasteleyn graph $g$, we prove +that it is good by induction on the number of edges it encloses. If it +is a minimal circuit (in particular if it encloses no edge), then it is +good by assumption. If not, then there exist $c_1$ and $c_2$ such that +$c=c_1\Delta c_2$, from which we conclude by the inductive hypothesis and +lemma~\ref{lemmacombcircuits}.\hfill$\square$ +\bigskip + +\theo{Lemma}\label{lemmakasteleynrmedge} +Given a Kasteleyn graph $g$, the graph obtained by removing any edge of $g$ +is Kasteleyn. +\endtheo +\medskip + +\indent\underline{Proof}: The lemma follows from lemma~\ref{lemmakasteleyncomplete} and the fact that minimal circuits of the graph obtained by removing the edge are circuits of $g$.\hfill$\square$ +\bigskip + +\indent We will now prove proposition~\ref{propdirectkasteleyn} and, in the process, provide an algorithm to direct a planar graph $g$ in such a way that the resulting directed graph is Kasteleyn. +\medskip + +{\bf Proof of proposition~\ref{propdirectkasteleyn}}: First of all, we notice that we can safely assume that $g$ has a boundary circuit: if it did not, then we construct an auxiliary graph $\gamma$ by adding edges to $g$, as detailed in section~\ref{secboundarycircuit}. Once $\gamma$ has been directed, the extra edges can be removed, and the Kasteleyn nature of the resulting directed graph then follows from lemma~\ref{lemmakasteleynrmedge}. + +\indent Assuming $g$ has a boundary circuit, we prove the proposition by induction on the number of +internal edges of the graph. + +\indent We first direct the edges of $\partial g$ in such a way that it is good +(if those edges are already directed then $\partial g$ is good by +assumption). + +\indent We first consider the case in which $\partial g$ is not a minimal circuit, +in which case there exist $c_1$ and $c_2$ such that $c_1\Delta c_2=c$. We +split $g$ into the graph $g_1$ consisting of $c_1$ and its interior and +$g_2$ consisting of $c_2$ and its interior. We direct the common edges +between $g_1$ +and $g_2$ (or equivalently between $c_1$ and $c_2$) in such a +way that $c_1$ is good (there may be many ways of doing so, any one will do). By the inductive hypothesis, this implies that +$g_1$ can be directed appropriately. By lemma~\ref{lemmacombcircuits}, $c_2$ +is good, which implies that $g_2$ can be directed as well. + +\indent We now turn to the case in which $\partial g$ is a minimal circuit (which +includes the case in which it has no interior edges). + +\indent If $\partial g$ encloses no circuit (that is if among the edges $\partial g$ +encloses, if any, none form a circuit), then none of the edges enclosed in +$\partial g$ belong to a minimal circuit of $g$ (since that circuit would +have to contain an edge of $\partial g$). Therefore the edges enclosed in +$\partial g$ can be directed in any way without affecting the Kasteleyn +nature of $g$. + +\indent If $\partial g$ encloses at least one circuit, let $c_1,\cdots,c_n$ be the +maximal circuits enclosed by $\partial g$. The edges that are outside all of +the $c_i$'s do not belong to any minimal circuit and can therefore be +directed in any way. Let $g_i$ be the sub-graph of $g$ consisting of $c_i$ +and its +interior. The sub-graph $g_i$ can be directed by the inductive +hypothesis.\hfill$\square$ +\bigskip + +\indent We will now discuss the notion of a {\it contraction} of a graph along an edge. +Given a Kasteleyn graph $g\in\mathcal G$ and two vertices $v_0,v_1$ that are connected by an edge that is directed from $v_0$ to $v_1$ ($v_0\succ v_1$), we define the \defd{contraction} $X_{v_0,v_1}g$ of $g$ over $(v_0,v_1)$ in the following way. We denote the family of vertices connected to $v_0$ and $v_1$ respectively, ordered in the {\it clockwise} direction, by $(v_1,u_1,\cdots,u_n)$ and $(v_0,u'_1,\cdots,u'_{n'})$. Let $(u'_{i_1},\cdots,u'_{i_m})\equiv +(u_1'',\ldots,u_m'')$ denote the ordered family obtained from $(u'_1,\cdots,u'_n)$ by removing the vertices that are in $(u_1,\cdots,u_n)$. The contraction $X_{v_0,v_1}g$ is constructed from $g$ by replacing $v_0$ and $v_1$ by a single vertex, which we also denote by $v_0$, connected to $(u_1,\cdots,u_n,u''_{1},\cdots,u''_{m})$ in the {\it clockwise} direction. In other words, the contraction of $g$ is obtained by {\it merging} $v_0$ and $v_1$, contracting the edge $\{v_0,v_1\}$, and removing the resulting double edges. Note that if $v_0\prec v_1$ we do not define the contraction operation $X_{v_0,v_1}g$ at all. + +{\bf Remark}: Strictly speaking, the graph $X_{v_0,v_1}g$ has an odd number of vertices and is therefore not an element of $\mathcal G$. This does not matter, since in the following we will always consider an even number of contractions. In the lemma below, the definition of a Kasteleyn graph with an odd number of vertices is identical to definition~\ref{defkasteleyn}. +\bigskip + +\theo{Lemma}\label{lemmacontraction} +Consider a Kasteleyn graph $g$ with a boundary circuit $\partial g$ and two vertices $v_0\succ v_1\in\mathcal V(\partial g)$ that are such that $\{v_0,v_1\}\in\mathcal E(\partial g)$, and $v_0$ precedes $v_1$ in the counterclockwise direction. The contraction $X_{v_0,v_1}g$ is Kasteleyn. +\endtheo +\medskip + +\indent\underline{Proof}: Using the notation introduced above, consider a circuit $c$ of $X_{v_0,v_1}g$. + +\indent If $c$ does not contain $v_0$ or contains $(u_i,v_0,u_j)$ for some $i\neq j$, then it is also a circuit of $g$, and is therefore oddly-directed in $X_{v_0,v_1}g$ if and only if it is oddly-directed in $g$. Since $\{v_0,v_1\}$ is in $\partial g$, the number of vertices encircled by $c$ is the same in $g$ and in +$X_{v_0,v_1}g$, which implies that $c$ is good. + +\indent If $c$ contains $(u''_i,v_0,u''_j)$ for some $i\neq j$, then we define a circuit $c'$ of $g$ by replacing $v_0$ by $v_1$. Again, +$c$ is oddly-directed in $X_{v_0,v_1}g$ if and only if $c'$ is oddly-directed in $g$. Since the number of vertices encircled by $c$ is the same as that by $c'$, we conclude that $c$ is good. + +\indent If $c$ contains $(u_i,v_0,u''_j)$ for some $i,j$, then we define a circuit $c'$ of $g$ by replacing $v_0$ by $(v_0,v_1)$. Since $v_0\succ v_1$, $c'$ is oddly-directed if and only if $c$ is oddly-directed which implies, noting again that $c$ and $c'$ encircle the same number of vertices, +that $c$ is good. + +\indent Since $v_0$ precedes $v_1$ in the counterclockwise direction and $c$ is a counterclockwise circuit, $c$ cannot contain $(u''_i,v_0,u_j)$ for any $i,j$.\hfill$\square$ +\bigskip + +{\bf Remark}: The requirement that $\{v_0,v_1\}$ is on the boundary circuit of $g$ is essential: indeed, if it were not so, one of the following would occur. If $v_0$ and $v_1$ were not both on the boundary, then the number of vertices enclosed by the boundary circuit would change parity when the vertices $v_0$ and $v_1$ were merged, and therefore the boundary circuit would no longer be good. +If $v_0$ and $v_1$ were both on the boundary but $\{v_0,v_1\}\not\in \mathcal E(\partial g)$, +then the edge $\{v_0,v_1\}$ would be part of {\it two} circuits, and would appear in one of them as $(u_i,v_0,v_1,u'_j)$ and as $(u'_j,v_1,v_0,u_i)$ in the other. Upon merging $v_0$ and $v_1$, one of the resulting circuits would be good while the other would not. + +\subsection{Extended dimer coverings}\label{subsecextendeddimer} + +\indent Proving the positivity of a Kasteleyn graph can be difficult: indeed, +one first needs to find a dimer covering of the graph, which is not always +easy (or possible, see figure~\ref{fignoncoverable}). In order to facilitate this task, we define the set of {\it +extended dimer coverings}, show how to construct them for any graph whose connected components have an even number of vertices, and +most importantly, show how to prove the positivity of a graph from one of +its extended dimer coverings. +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/noncoverable.pdf}\par\penalty10000 +\medskip +\captionshort{A graph that admits no dimer covering.} +\label{fignoncoverable} +\end{figure} + +\indent An \defd{extended dimer covering} of a graph $g$ is a collection of dimers +such that every vertex is covered by an {\it odd} number of dimers (not necessarily by one dimer as is the case for usual dimer +coverings). +See figure~\ref{figextendedcover} for an example. +\bigskip + +\begin{figure} +\hfil\includegraphics[width=8cm]{Figs/extended_dimer.pdf}\par\penalty10000 +\medskip +\caption{An extended dimer covering: $(1,6)$, $(2,8B)$, $(3,8A)$, $(4,5A)$, $(5C,8C)$, $(7,5B)$. The covering is positive.} +\label{figextendedcover}\end{figure} + +\indent One of the nice features of extended dimer coverings is that any connected +graph with an even number of vertices admits at least one extended dimer +covering. This fact is proved in the following proposition, and an algorithm +for constructing an extended dimer covering of a graph is provided in +its proof. +\bigskip + +\theo{Proposition}\label{propcovextended} +Given a connected graph $g\in\mathcal G$ (recall that the graphs in $\mathcal G$ have an even number of vertices), $g$ admits at least one extended dimer covering. +\endtheo +\medskip + +\indent\underline{Proof}: Let $\mathcal T$ denote a {\it spanning tree} of $g$, +that is a connected sub-graph of $g$ that covers all the vertices of $g$ and +has no loops. We then construct an extended dimer covering of $\mathcal T$, +which, obviously, is also an extended dimer covering of $g$. + +\indent We fix one of the vertices of $g$ as the root of $\mathcal T$. We denote the root of $\mathcal T$ by +$v_0$. We construct an extended dimer covering by induction on the {\it +height} of the tree (the height of a vertex $v$ is the number of edges one +needs to traverse to get from $v$ to $v_0$, and the height of a tree is the maximal height of its vertices). +If the height of $\mathcal +T$ is 1, then we obtain the desired extended dimer covering of $\mathcal T$ +by putting a dimer on every edge of $\mathcal T$. If the height $h$ of $\mathcal +T$ is larger than $1$, +we put a dimer on every edge between a vertex of height $h-1$ and a vertex of height $h$. +We will call these dimers {\it terminal dimers}. +At this point, we let $\mathcal T'$ be the tree obtained by removing from $\mathcal T$ all +the vertices of height $h$ as well as the vertices of height $h-1$ that are touched by an odd number of dimers. +Note that the height of $\mathcal T'$ is $\le h-1$. By the inductive hypothesis, there exists an extended dimer covering $\sigma'$ of +$\mathcal T'$. By appending the terminal dimers to $\sigma'$, we obtain an extended dimer covering of $\mathcal T$. +\hfill$\square$ +\bigskip + +\indent We will now show how to prove positivity from extended dimer coverings.\par +\smallskip +Similarly to dimer coverings, we associate a permutation +$\pi_{\bar\sigma}^{(\omega)}$ to each extended dimer covering $\bar\sigma$. +We write +\begin{equation} +\bar\sigma=\{(v_1,v_2),\cdots,(v_{|\bar\sigma|-1},v_{|\bar\sigma|})\} +\label{eqbarsigma}\end{equation} +such that $v_{2i-1}\succ v_{2i}$. In this expression the $v_i$'s are +not +necessarily different though they can only appear an {\it odd} number of +times. In addition +to the labeling $\omega(v_i)$, we associate a {\it +letter} $\alpha(v_i)\in\{A,B,\cdots\}$ to $v_i$, in such a way +that when more than one dimer is attached to $v_i$, the letters associated +to these dimers are ordered alphabetically in $\bar\sigma$ in the {\it clockwise} direction. +The permutation +$\pi_{\bar\sigma}^{(\omega)}\in\mathcal S_{|\bar\sigma|}$ is the +permutation that orders $(v_1,\cdots,v_{|\bar\sigma|})$ in {\it +lexicographical order} (e.g. $1A<2A<2B<2C<3A$). The extended dimer +covering $\bar\sigma$ is said to be \defd{positive} if the signature of +$\pi_{\bar\sigma}^{(\omega)}$ is positive. The extended dimer covering in the example in figure~\ref{figextendedcover} is positive. + +\indent The analog of proposition~\ref{propkasteleyn} holds for extended dimer +coverings. +\bigskip + +\theoname{Proposition}{Uniform positivity for extended dimer coverings}\label{propextendedkasteleyn} + Given a vertex labeling $\omega$, a Kasteleyn graph $g$ is positive if and +only if one of its extended dimer coverings is positive. +\endtheo +\medskip + +\indent\underline{Proof}: The main idea of the proof is to introduce an auxiliary +graph $\gamma$ by adding edges and vertices to $g$ such that every extended +dimer covering of $g$ corresponds uniquely to a usual dimer covering of $\gamma$ that has the same sign $s\in\{-1,1\}$. In addition, $\gamma$ is constructed in +such a way that it is Kasteleyn, which implies that the sign of all of its dimer +coverings is $s$. In particular the sign of the coverings that correspond to +usual dimer coverings of $g$ is $s$ as well. +\medskip + +\indent The auxiliary graph $\gamma$ is constructed from $g$ by replacing every +vertex of $g$ by a collection of triangles as in figure~\ref{figtriangles} (see figure~\ref{figextendedcovertriangle} for an example). +It is straightforward to check that there is a one-to-one correspondence between the extended dimer +coverings of a single vertex and the dimer coverings of the +corresponding triangle-replacement (see figure~\ref{figtrianglescovbi}). +This induces a one-to-one mapping between the dimer coverings of +$\gamma$ and the extended dimer coverings of $g$. +\bigskip + +\begin{figure} +\hfil +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3_pre.pdf}} +\hskip20pt$\longmapsto$\hskip20pt +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3.pdf}}\hskip20pt +\hfil +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_4_pre.pdf}} +\hskip20pt$\longmapsto$\hskip20pt +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_4.pdf}}\par\penalty10000 +\bigskip +\hfil +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_5_pre.pdf}} +\hskip20pt$\longmapsto$\hskip20pt +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_5.pdf}}\par\penalty10000 +\bigskip +\caption{Replacing vertices with triangles. Only vertices with 3, 4 and 5 +edges are drawn, the other cases are treated similarly.} +\label{figtriangles} +\end{figure} +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/extended_dimer_triangle.pdf}\par\penalty10000 +\medskip +\caption{The auxiliary graph $\gamma$ corresponding to the graph in figure~\ref{figextendedcover}. The dimer covering associated to the extended dimer covering in figure~\ref{figextendedcover} is represented as well. Its labels are $(1,6)$, $(2,8d)$, $(3,8b)$, $(8c,8a)$, $(4a,4b)$, $(4c,5a)$, $(5e,5c)$, $(5d,8g)$, $(8e,8f)$, $(7,5b)$.} +\label{figextendedcovertriangle} +\end{figure} + +\begin{figure} +\hfil +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3_Xcover1.pdf}} +\hskip20pt$\longmapsto$\hskip20pt +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3_cover1.pdf}}\hskip20pt +\hfil +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3_Xcover3.pdf}} +\hskip20pt$\longmapsto$\hskip20pt +\parbox[m]{2cm}{\includegraphics[width=2cm]{Figs/triangle_3_cover3.pdf}}\par\penalty10000 +\bigskip +\caption{Bijection between the set of extended dimer coverings of a vertex and the set of dimer coverings of its triangle-replacement. The case with 3 external lines is drawn, the others are treated similarly.} +\label{figtrianglescovbi} +\end{figure} + + +\indent The extra triangles in $\gamma$ are directed as per +figure~\ref{figtriangledirect}. We now show that $\gamma$ is Kasteleyn. Each +triangle is a minimal circuit of $\gamma$ and is good. In any other +minimal circuit containing an edge of a triangle, that edge will be {\it +forwards}, which implies that the +circuit is good if and only if the corresponding circuit in $g$ is +good as well. This proves that $\gamma$ is Kasteleyn. +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/triangle_5_directed.pdf}\par\penalty10000 +\medskip +\caption{The direction and labeling of the extra triangles in $\gamma$. The +example for 5 external edges is shown, the others are treated similarly.} +\label{figtriangledirect} +\end{figure} + + +\indent Finally, we show that an extended dimer covering of $g$ is positive if and only if the +corresponding dimer covering of $\gamma$ is positive. To do so, we label +the vertices of $\gamma$ in the following way. In order to simplify the +discussion, the label of a vertex in $\gamma$ will be a number and a letter +instead of just a number. Given a vertex $v$ in $g$ labeled by $\omega(v)$, +the number in the label of each vertex of the corresponding +triangle-replacement in $\gamma$ is set to $\omega(v)$. The letters of the +labels of the vertices in each triangle-replacement are assigned as per +figure~\ref{figtriangledirect}. Note that the external lines in +figure~\ref{figtriangledirect} are labeled alphabetically in the {\it +clockwise} direction, as were the dimers in~(\ref{eqbarsigma}). + +\indent Consider a dimer covering $\sigma$ of $\gamma$ and its associated +extended dimer covering $\bar\sigma$ of $g$. There are two types of dimers +in $\sigma$: {\it internal} dimers which cover an edge of one of the extra +triangles, and {\it external} dimers. Internal dimers are labeled as +$(x\alpha,x\alpha')$ with $x\alpha\succ x\alpha'$, where $x\in\mathbb N$ is the number-label corresponding +to the triangle that contains it and $\alpha,\alpha'\in\{a,b,\cdots\}$ are +the letter-labels of its extremities, while external dimers are labeled as $(x\alpha,y\alpha')$ with $x\alpha\succ y\alpha'$, where $x\neq y$ are the number-labels corresponding to the triangles that its extremities are connected to and $\alpha,\alpha'$ are the letter-labels corresponding to its extremities. + +\indent We write the dimer covering $\sigma$ as an ordered collection of labels +$$ +\sigma=(x_1\alpha_1,x_2\alpha_2,\cdots,x_{|\gamma|}\alpha_{|\gamma|}) +$$ +in which the dimers of $\sigma$ are $(x_{2i-1}\alpha_{2i-1},x_{2i}\alpha_{2i})$. Similarly, we write $\bar\sigma$ as +$$ +\bar\sigma=(y_1\beta_1,y_2\beta_2,\cdots,y_{|\bar\sigma|}\beta_{|\bar\sigma|}). +$$ +With no loss of generality, we assume that the labels of the vertices covered by external dimers +appear in the same order both in $\sigma$ and in $\bar\sigma$. + +\indent The claim of the proposition is that $\sigma$ can be ordered by a positive-signature permutation if and only if $\bar\sigma$ can be. In order to prove this, we will {\it group} some labels in $\sigma$ together and label the groups in such a way that $\sigma$, written in terms of its groups, has a similar expression as $\bar\sigma$. + +\indent We denote the set of labels of vertices covered by external dimers of $\sigma$ by $\epsilon(\sigma)$. Given a +number-label $x$, we let $\iota_x(\sigma)=\{\beta\,:\,x\beta$ is covered by an internal dimer of $\sigma\}$. +In addition, given one of the extra triangles in $\gamma$ and two of its vertices, with letter-labels $\beta,\beta'$, we denote the letter-label of the third vertex of the triangle by $\tau_{\beta,\beta'}$ (e.g., $\tau_{b,c}=a$ and $\tau_{c,e}=d$, see figure~\ref{figtriangledirect}). +For every $x\alpha\in \epsilon(\sigma)$ we define an ordered set +$x\bar\alpha$ containing $x\alpha$ and the labels $x\beta\in\iota_x(\sigma)$, which are such that, +if $(x\beta,x\beta')$ is a dimer and $x\beta,x\beta'\in x\bar\alpha$, then $x\tau_{\beta,\beta'}\in x\bar\alpha$. The ordering of $x\bar\alpha$ +is that inherited from $\sigma$. The set $x\bar\alpha$ can either be {\it trivial}, if it has only one element (in which case $x\bar\alpha=(x\alpha)$), +or it can be of three possible non-trivial types, depicted in figure~\ref{fig:4.7}. + +\indent In all cases, $x\bar\alpha$ is, up to a positive-signature permutation, +of the form +$$x\bar\alpha=(x[\alpha-k],\ldots,x\alpha,\ldots, x[\alpha+k']),$$ +where $k,k'\ge 0$, $k+k'$ is even and $[\alpha+i]$ is the character-wise addition (e.g., $[a+2]=c$ and $[j-6]=d$). +\bigskip + +\begin{figure} +\hfil\includegraphics[width=8cm]{Figs/triangle_groups_lr.pdf}\par\penalty10000 +\bigskip +\hfil\includegraphics[width=6cm]{Figs/triangle_groups_l.pdf} +\hfil\includegraphics[width=6cm]{Figs/triangle_groups_r.pdf}\par\penalty10000 +\bigskip +\caption{The three possible types of non-trivial sets $x\bar\alpha$, and their corresponding dimer covering. In each of these, the labels of the vertices covered by dimers can be written as $(x[\alpha-k],\ldots,x\alpha,\ldots, x[\alpha+k'])$. In the first case, $k,k'$ are both odd. In the second case, +$k$ is positive and even, while $k'=0$. In the third case, $k=0$, while $k'$ is positive and even. +In addition to these types, $x\bar\alpha$ could be trivial, that is $k=k'=0$.} +\label{fig:4.7} +\end{figure} + +\indent In addition, it is straightforward to check that, by a permutation that rigidly moves the labels of pairs of vertices covered by the same dimer, $\sigma$ can be turned into +$$ +\sigma'\equiv(y_1\bar\alpha_1,\cdots,y_{|\bar\sigma|}\bar\alpha_{|\bar\sigma|}) +$$ +where the $y_i$'s are the same, and are ordered in the same way, as in $\bar\sigma$. +Note that, since the sets $y_i\bar\alpha_i$ have odd cardinality and contain ordered sequences of labels, the permutation that orders $\sigma$ and the permutation that orders $\sigma'$ have the same signature. We are therefore left with proving that this permutation has the same signature as that which orders $\bar\sigma$. + +\indent The indices $\alpha_i$ associated with the indices $\bar\alpha_i$ in $\sigma'$ are not equal, in general, to the indices $\beta_i$ in $\bar\sigma$. However, +they appear in the same order as the $\beta_i$'s, i.e. if $y\bar\alpha_i$ and $y\bar\alpha_j$ (resp. +$y\beta_i$ and $y\beta_j$) appear in $\sigma'$ (resp. $\bar\sigma$) in that order, +then $\alpha_i<\alpha_j$ if and only if $\beta_i<\beta_j$. Indeed, restricting our attention to the elements that share the same number-label $y$: +$$ +y\bar\alpha_{i_1},\cdots,y\bar\alpha_{i_k}\qquad\mathrm{and}\qquad +y\beta_{i_1},\cdots,y\beta_{i_k} +$$ +we notice that the permutations that order these sequences in the lexicographic order, +also order the vertices in the +clockwise direction. Therefore, the $y\bar\alpha_i$'s can be ordered by the same permutation as the $y\beta_i$'s, which concludes the proof of the proposition.\hfill$\square$ + + +\section{Evenly filled graphs}\label{seceven} + +\indent We will now delve into the next level of complexity and consider graphs +with interior edges and vertices. For the time being, we require that every +connected component of the graph formed by the interior edges and vertices +has an even number of vertices. The general case will be discussed in the +next section. +\bigskip + +\indent The family of \defd{evenly filled graphs} $\mathcal G_e\subset\mathcal G$ is +defined +in the following way. Evenly filled graphs are connected and consist of an +interior +graph $\check g\in\mathcal G$ each of whose connected components contains an +even number of vertices, enclosed +within a boundary circuit of vertices $\partial g$. Since evenly filled graphs are +connected, $\partial g$ must be connected to each connected component of +$\check g$ by at least one +edge. In addition, vertices of $\partial g$ may be connected to each other +by internal edges. See figure~\ref{figevenexample} for an example. Note that $\mathcal G_s\subset\mathcal G_e\subset\mathcal G$. +\hugeskip + +\begin{figure} +\hfil\includegraphics{Figs/even_example.pdf}\par\penalty10000 +\medskip +\captionshort{An evenly filled graph.} +\label{figevenexample} +\end{figure} + +\indent We will now describe how to direct and label an evenly filled graph in such a way that its boundary MD partition function can be written as a Pfaffian as in theorem~\ref{theomain}. +\bigskip + +\delimtitleref{Directing and labeling an evenly filled graph}\label{recipeeven} +We label the vertices of $\partial g$ by following the edges of $\partial g$ sequentially in the counterclockwise direction. The resulting labeling is denoted by $\omega$. The location of +the vertex labeled as 1 is unimportant. + +\indent We then direct the edges of $\partial g$: +$\omega^{-1}(i)\succ\omega^{-1}(i+1)$ and +$\omega^{-1}(1)\succ\omega^{-1}(|\partial g|)$. This implies that +$\partial g$ is good so that the remaining edges of +$g$ can be directed by +proposition~\ref{propdirectkasteleyn}. We arbitrarily choose one of the directions constructed in its proof, see section \ref{subsecpropkasteleyn}. + +\indent Finally, we label the remaining vertices: we consider an extended dimer +covering of $\check g$, which exists by proposition~\ref{propcovextended}, +and set the labels in such a way that this covering is positive. In order to +construct such a labeling, one can pick a random labeling of the vertices, +check whether the extended dimer covering is positive, and exchange two +labels if it is not. +\enddelim +\bigskip +See figure~\ref{figevenexamplelabels} for an example. +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/even_example_label_pre.pdf} +\hfil\includegraphics{Figs/even_example_label.pdf}\par\penalty10000 +\medskip +\caption{Two stages of the directing and labeling of the graph in figure~\ref{figevenexample}. The first represents $g$ after having labeled $\partial g$. The second shows an extended dimer covering (which happens to be a usual dimer covering) of $\check g$ and an associated labeling. The labels are chosen in such a way that this covering is positive: $(13,14)$, $(15,16)$.} +\label{figevenexamplelabels} +\end{figure} + +\indent Thus directed, the graph $g$ is Kasteleyn. We now prove that it is positive by constructing an extended dimer covering of the entire graph $g$ by combining the extended dimer covering of $\check g$ mentioned above and the boundary-dimers +$$ +\{(\omega^{-1}(1),\omega^{-1}(2)),\cdots,(\omega^{-1}(|\partial g|-1),\omega^{-1}(|\partial g|))\}. +$$ +Since the boundary-dimers are labeled sequentially, this covering is positive, which implies that $g$ is positive as well. See figure~\ref{figevenexamplecover} for an example. +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/even_example_cover.pdf}\par\penalty10000 +\medskip +\caption{An extended dimer covering of the graph in figure~\ref{figevenexample} with the labeling of figure +\ref{figevenexamplelabels} +(which happens to be a usual dimer covering). The covering can be labeled as +$(1,2)$, $(3,4)$, $(5,6)$, $(7,8)$, $(9,10)$, $(11,12)$, $(13,14)$ and its associated permutation is the identity.} +\label{figevenexamplecover} +\end{figure} + +\indent If the monomers of an MD covering are fixed on vertices of $\partial +g$, the possible dimer +positions are the possible (pure) dimer coverings of the sub-graph of $g$ +obtained by removing the vertices that have a monomer. We will now prove +that such a sub-graph is Kasteleyn and positive which implies that the dimer +partition function on it satisfies the Pfaffian +formula~(\ref{eqkasteleyntheo}) which can be substituted +in~(\ref{eqliebtheo}) to reproduce the MD partition function. + +\indent Given a family +of monomers $\mathcal M\subset\mathcal V(\partial g)$ of even cardinality, +we +define $[g]_{\mathcal M}$ as the sub-graph of $g$ obtained by removing the +vertices in $\mathcal M$, and define a new labeling $\omega_{\mathcal M}$ +for its vertices that is obtained from $\omega$ by removing the monomeric +vertices and eliminating the labeling gaps: +\begin{equation} +\omega_{\mathcal M}(v):=\omega(v)-\mathrm{card}\{v'\in\mathcal M\quad|\quad +\omega(v')<\omega(v)\}. +\label{eqinducedlabelingeven}\end{equation} +See figure~\ref{figevenexamplemonomer} for an example. Note that this definition is ill-posed if $[g]_{\mathcal M}$ is empty (i.e. if $g$ does not have interior vertices and $\mathcal M=\mathcal V(g)$), but this is of little consequence. + +\begin{figure} +\hfil\includegraphics{Figs/even_example_monomer.pdf} +\hfil\includegraphics{Figs/even_example_auxiliary.pdf}\par\penalty10000 +\medskip +\caption{The sub-graph $[g]_{\mathcal M}$ and the auxiliary graph $\gamma_{\mathcal M}$ for the graph in +figure~\ref{figevenexamplelabels} with +monomers on the vertices labeled by 1, 8, 10 and 12.} +\label{figevenexamplemonomer}\end{figure} +\bigskip + +\theo{Lemma}\label{lemmaevenpositivity} +For every $g\in\mathcal G_e$, directed and labeled as above, for all +$\mathcal M\subset\mathcal V(\partial g)$ of even cardinality $|\mathcal +M|$, the sub-graph +$[g]_{\mathcal M}$ whose vertices are labeled by $\omega_{\mathcal M}$ is +Kasteleyn and positive. +\endtheo +\medskip + +\indent\underline{Proof}: We first notice that $[g]_{\mathcal M}$ is Kasteleyn +because, since the monomers are on $\partial g$, the minimal circuits of +$[g]_{\mathcal M}$ are minimal circuits of $g$. In order to prove its +positivity, we construct an auxiliary graph +$\gamma_{\mathcal M}$, which is a super-graph of $[g]_{\mathcal M}$ (i.e. +$[g]_{\mathcal M}$ is a sub-graph of $\gamma_{\mathcal M}$), by adding extra +edges in such a way that $\gamma_{\mathcal M}$ is Kasteleyn, and that its +positivity is obvious. The positivity of +$[g]_{\mathcal M}$ then follows from proposition~\ref{propkasteleyn}. + +\indent In order to define $\gamma_{\mathcal M}$, we first define the sub-graph $\eta_{\mathcal M}$ of $g$ obtained by removing the edges that are connected to vertices in $\mathcal M$ and are not in $\mathcal E(\partial g)$. In other words, $\eta_{\mathcal M}$ is the union of $[g]_{\mathcal M}$ and $\partial g$. +It follows from lemma~\ref{lemmakasteleynrmedge} that $\eta_{\mathcal M}$ is Kasteleyn. For every vertex $v\in\mathcal M$, we define the vertex $u(v)$ in the following way: +\begin{itemize} +\item if the set $\{\omega(v')\ |\ v'\in\mathcal V(\partial g)\setminus\mathcal M,\ \omega(v')<\omega(v)\}$ is not empty, then $\omega(u(v))$ is its maximizer. +\item if the set is empty, then $\omega(u(v))$ is the minimizer of $\{\omega(v')\ |\ v'\in\mathcal V(\partial g)\setminus\mathcal M,\ \omega(v')>\omega(v)\}$. +\end{itemize} +Note that $u(v)$ is separated from $v$ on $\partial g$ only by vertices in $\mathcal M$ and edges that are $\neq\{\omega^{-1}(1),$ $\omega^{-1}(|\partial g|)\}$. +Note also that in general there might be $v_1\neq v_2$ such that $u(v_1)=u(v_2)$. Given this notion, +$\gamma_{\mathcal M}$ is defined as the {\it contraction} $$\big(\prod_{\mAthop{v\in\mathcal M}_{u(v)\prec v}}X_{u(v),v} +\prod_{\mAthop{v\in\mathcal M}_{u(v)\succ v}}X_{v,u(v)}\big)\eta_{\mathcal M},$$ +where the $X_{u(v),v}$ should be multiplied in an order such that the product is well defined (see figure~\ref{figevenexamplemonomer} for an example). + +\indent Thus defined, $\gamma_{\mathcal M}$ is a super-graph of $[g]_{\mathcal M}$, and, because of the way $\omega$ and $u(v)$ were defined, the edges that are contracted are all directed in the {\it counterclockwise} direction, which allows us to apply lemma~\ref{lemmacontraction} to prove that $\gamma_{\mathcal M}$ is Kasteleyn. + +\indent We are left with proving that $\gamma_{\mathcal M}$ is positive. Because of the way the labeling $\omega_{\mathcal M}$ was defined, the dimer +covering corresponding to the identity-permutation restricted to the boundary of $\gamma_{\mathcal M}$ is an acceptable dimer covering. The interior of $\gamma_{\mathcal M}$ is the same as the interior of $g$, and can therefore be covered by the same extended dimer covering as $\check g$. Therefore, $\gamma_{\mathcal M}$ is positive. See figure~\ref{figevenexampleauxcover} for an example.\hfill$\square$ +\bigskip + +\begin{figure} +\hfil\includegraphics{Figs/even_example_auxcover.pdf}\par\penalty10000 +\medskip +\caption{An extended dimer covering of $\gamma_{\mathcal M}$ (which happens to be a usual dimer covering).} +\label{figevenexampleauxcover} +\end{figure} + +\indent It follows from lemma~\ref{lemmaevenpositivity} and +theorem~\ref{theokasteleyn} that by defining $a(\mathbf d)$ as in~(\ref{eqAdef}), +$\mathrm{pf}([a(\mathbf d)]_{\mathcal M})$ is the partition function of dimers on $[g]_{\mathcal M}$, +which, by theorem~\ref{theolieb}, implies +theorem \ref{theomain} for evenly filled graphs, provided $g\in\mathcal G_e$ is directed and labeled as above (see ``\ref{recipeeven}''). + + +\section{Enclosed graphs}\label{secenclosed} + +\indent We now turn to the more general case of graphs that have a boundary circuit. +\bigskip + +\indent The family of \defd{enclosed graphs} $\mathcal G_{en}\subset\mathcal G$ is +defined +in the following way. Enclosed graphs are connected and consist of an +interior +graph $\check g\in\mathcal G$ enclosed +within a boundary circuit of vertices $\partial g$ containing an {\it even} number $|\partial g|$ of vertices. Since enclosed graphs are +connected, $\partial g$ must be connected to each connected component of +$\check g$ by at least one +edge. In addition, vertices of $\partial g$ may be connected to each other +by internal edges. See figure~\ref{figenclosedexample} for an example. Note that since $|g|$ is even, $|\check g|$ is even as well. Notice that $\mathcal G_s\subset\mathcal G_e\subset\mathcal G_{en}\subset\mathcal G$. +\bigskip + +\begin{figure} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example.pdf}\par\penalty10000 +\medskip +\captionshort{An enclosed graph.} +\label{figenclosedexample}\end{figure} + +\indent We will now describe how to direct and label an enclosed graph in such a way that its boundary MD partition function can be written as a Pfaffian as in theorem~\ref{theomain}. See figure~\ref{figenclosedexamplecover} for an example. +\bigskip + +\delimtitleref{Directing and labeling an enclosed graph}\label{recipeenclosed} +We first label the vertices of $\partial g$ following the edges of $\partial g$ sequentially in the counterclockwise direction. The resulting labeling is denoted by $\omega$. The location of the vertex labeled as 1 is unimportant. + +\indent We then direct the edges of $\partial g$: +$\omega^{-1}(i)\succ\omega^{-1}(i+1)$ and $\omega^{-1}(1)\succ\omega^{-1}(|\partial g|)$. +This implies that $\partial g$ is good. The +remaining edges of $g$ can be directed by +proposition~\ref{propdirectkasteleyn}. We arbitrarily choose one of the directions constructed in its proof, see section \ref{subsecpropkasteleyn}. + +\indent Finally, we label the remaining vertices by constructing an extended dimer +covering of $g$ and requiring that it +be positive. By +proposition~\ref{propcovextended}, such a covering exists. We choose $\omega$ by picking a random labeling of the vertices, checking whether the extended dimer +covering is positive, and exchanging two labels if it is not. +\enddelim +\bigskip + +\begin{figure} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_directed.pdf} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_cover.pdf}\par\penalty10000 +\medskip +\caption{Two stages of the directing and labeling of the graph in figure~\ref{figenclosedexample}. The first represents $g$ after having labeled the vertices of $\partial g$. The second shows an extended dimer covering of $g$ and an associated labeling. The labels are chosen in such a way that this covering is positive: +$(1,2)$, $(3,4B)$, $(4A,5)$, $(7,8A)$, $(8B,4C)$, $(10,8C)$ $(6,9)$, $(11,12)$, $(13,14A)$, $(14B,15)$, $(16,14C)$.} +\label{figenclosedexamplecover}\end{figure} + + +\indent If the monomers of an MD covering are fixed on vertices of $\partial +g$, the possible dimer +positions are the possible (pure) dimer coverings of the sub-graph of $g$ +obtained by removing the vertices that have a monomer. We will now prove +that such a sub-graph is Kasteleyn and positive which implies that the dimer +partition function on it satisfies the Pfaffian +formula~(\ref{eqkasteleyntheo}) which can be substituted +in~(\ref{eqliebtheo}) to reproduce the MD partition function. + +\indent Given a family +of monomers $\mathcal M\subset\mathcal V(\partial g)$ of even cardinality, +we +define $[g]_{\mathcal M}$ as the sub-graph of $g$ obtained by removing the +vertices in $\mathcal M$, and define a new labeling $\omega_{\mathcal M}$ +for its vertices that is obtained from $\omega$ by removing the monomeric +vertices and eliminating the labeling gaps: +\begin{equation} +\omega_{\mathcal M}(v):=\omega(v)-\mathrm{card}\{v'\in\mathcal M\quad|\quad +\omega(v')<\omega(v)\}. +\label{eqinducedlabelingenclosed}\end{equation} +See figure~\ref{figetareplace2} for an example. +\bigskip + +\theo{Lemma}\label{lemmaenclosedpositivity} +For every $g\in\mathcal G_{en}$, directed and labeled as above, for all +$\mathcal M\subset\mathcal V(\partial g)$ of even cardinality $|\mathcal +M|$, the sub-graph +$[g]_{\mathcal M}$ whose vertices are labeled by $\omega_{\mathcal M}$ is +Kasteleyn and positive. +\endtheo +\medskip + +\indent\underline{Proof}: We first notice that $[g]_{\mathcal M}$ is Kasteleyn +because, since the monomers are on $\partial g$, the minimal circuits of +$[g]_{\mathcal M}$ are minimal circuits of $g$. In order to prove its +positivity, we will first construct an auxiliary graph $\varphi$ by adding edges and vertices to $g$, which is evenly filled. We will then show that $[g]_{\mathcal M}$ is positive if and only if $[\varphi]_{\mathcal M}$ is, by constructing an injective map $\lambda_{\varphi,\mathcal M}$ from the set of extended dimer coverings of $[g]_{\mathcal M}$ to those of $[\varphi]_{\mathcal M}$ that preserves positivity. +Finally, using the fact that $\varphi$ is evenly filled and lemma \ref{lemmaevenpositivity} we conclude that $[\varphi]_{\mathcal M}$ is positive. +\medskip + +\indent We construct $\varphi$ from $g$ in the following way. Consider $\check g$. We add as many edges as needed to connect the connected components of $\check g$ to each other. Some of these extra edges may cross edges of $g$. We replace every such crossing as per figure~\ref{figetareplace}. Note that by this construction, some edges may be crossed multiple times (this could be avoided by choosing the edges to add to $\check g$ carefully, but it is not necessary to do so). +\bigskip + +\begin{figure} +\hfil\parbox[m]{4cm}{\includegraphics[width=4cm]{Figs/etareplace_cross.pdf}} +\hfil$\longmapsto$ +\hfil\parbox[m]{4cm}{\includegraphics[width=4cm]{Figs/etareplace_r.pdf}}\par\penalty10000 +\medskip +\caption{Replacing a crossing in the construction of $\varphi$. The solid line is the edge that is in $\mathcal E(g)$, and the dotted line is the extra edge that was added to $\varphi$ in order to connect $\check g$.} +\label{figetareplace}\end{figure} + +\indent The edges of $\varphi$ are directed in the following way. First of all, the edges in $\mathcal E(\varphi)\cap\mathcal E(g)$ are directed in the same way as in $g$. Next, the edges in $\mathcal E(\varphi)\setminus\mathcal E(g)$ corresponding to the solid lines in figure~\ref{figetareplace} +are directed as shown there. The circuits containing such edges are good, since the parity of the number of backwards edges is unchanged by the replacement. Finally, we direct the remaining edges in $\varphi$ by using the proof proposition~\ref{propdirectkasteleyn}. + +\indent We introduce the following notation: given an edge $e=\{v,v'\}\in\mathcal E(g)\setminus\mathcal E(\varphi)$, that is, an edge of $g$ that was replaced when removing edge crossings, if $v\succ v'$, then we define $(v,$ $\chi_1(e),\cdots,\chi_{2s_e}(e),$ $v')$ as the ordered string of vertices that replaces $e$. + +\indent The vertex labeling $\omega_\varphi$ of $\varphi$ is defined in the following way. The vertices $v$ in $\mathcal V(\varphi)\cap\mathcal V(g)$ are labeled in the same way as in $g$: $\omega_\varphi(v):=\omega(v)$. The pairs $(\chi_{2i-1}(e),\chi_{2i}(e))$ of vertices that were added to $\varphi$ by replacing a crossing are labeled in such a way that $\omega_\varphi(\chi_{2i}(e))=\omega_\varphi(\chi_{2i-1}(e))+1$ and $\omega_\varphi(\chi_{2i-1}(e))>|g|$. +\medskip + +\indent We now construct an injective mapping $\lambda_{\varphi,\mathcal M}$ from the extended dimer coverings of $[g]_{\mathcal M}$ to those of $[\varphi]_{\mathcal M}$ and show that $\lambda_{\varphi,\mathcal M}$ preserves positivity. Given an extended dimer covering $\sigma$ of $[g]_{\mathcal M}$, we construct an extended dimer covering $\lambda_{\varphi,\mathcal M}(\sigma)$ of $[\varphi]_{\mathcal M}$ in the following way. We first add the dimers of $\sigma$ that occupy an edge that is in $\mathcal E([g]_{\mathcal M})\cap\mathcal E([\varphi]_{\mathcal M})$ to $\lambda_{\varphi,\mathcal M}(\sigma)$. +\begin{itemize} +\item If an edge $e=\{v,v'\}\in\mathcal E([g]_{\mathcal M})\setminus\mathcal E([\varphi]_{\mathcal M})$ with $v\succ v'$ is occupied by a dimer, then we add the dimers $\{v,\chi_{1}(e)\}$, $\{\chi_{2i}(e),\chi_{2i+1}(e)\}$ for $1\le i\le s_e-1$, $\{\chi_{2s_e}(e),v'\}$ to $\lambda_{\varphi,\mathcal M}(\sigma)$. +\item If an edge $e=\{v,v'\}\in\mathcal E([g]_{\mathcal M})\setminus\mathcal E([\varphi]_{\mathcal M})$ with $v\succ v'$ is not occupied by a dimer, then we add the dimers $\{\chi_{2i-1}(e),\chi_{2i}(e)\}$ for $1\le i\le s_e$, to $\lambda_{\varphi,\mathcal M}(\sigma)$. +\item Similarly, for every edge $e=\{v,v'\}\in\left(\mathcal E(g)\setminus\mathcal E(\varphi)\right)\setminus\mathcal E([g]_{\mathcal M})$ with $v\succ v'$ (i.e. every edge that is in $g$, not in $\varphi$, and has at least one endpoint in $\mathcal M$), we add the dimers $\{\chi_{2i-1}(e),\chi_{2i}(e)\}$ for $1\le i\le s_e$, to $\lambda_{\varphi,\mathcal M}(\sigma)$. +\end{itemize} +See figures~\ref{figetareplaceocc}, \ref{figetareplace1} and~\ref{figetareplace2} for examples. +\bigskip + +\begin{figure} +\hfil\includegraphics[width=4cm]{Figs/etareplace_occupied.pdf} +\hfil\includegraphics[width=4cm]{Figs/etareplace_empty.pdf}\par\penalty10000 +\medskip +\caption{Covering of the replacement edges of $\varphi$. Left: the corresponding edge in $g$ is occupied. Right: the corresponding edge in $g$ is not occupied.} +\label{figetareplaceocc}\end{figure} + +\begin{figure} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_auxiliary_cover.pdf}\par\penalty10000 +\medskip +\captionshort{The covering of the auxiliary graph $\varphi$ corresponding to the covering in figure~\ref{figenclosedexamplecover}.} +\label{figetareplace1}\end{figure} +\bigskip + +\begin{figure} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_monomer_cover.pdf} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_monomer_auxiliary_cover.pdf}\par\penalty10000 +\medskip +\caption{The sub-graph $[g]_{\{6,8\}}$, one of its extended dimer coverings, and the corresponding covering of $[\varphi]_{\{6,8\}}$.} +\label{figetareplace2}\end{figure} + +\indent We now prove that $\sigma$ is positive if and only if $\lambda_{\varphi,\mathcal M}(\sigma)$ is. We write $\sigma$ as an ordered sequence of vertices $(v_1,\cdots,v_{2n})$ in which the dimers occupy $\{v_{2i-1},v_{2i}\}$ with $v_{2i-1}\succ v_{2i}$. By construction, $\lambda_{\varphi,\mathcal M}(\sigma)$ is obtained from $\sigma$ by +\begin{itemize} +\item replacing $(v,v')$, in which $e=\{v,v'\}\in\mathcal E([g]_{\mathcal M})\setminus\mathcal E([\varphi]_{\mathcal M})$, by $(v,\chi_1(e),$ $\cdots,\chi_{2s_e}(e),v')$, +\item adding $(\chi_1(e),\cdots,\chi_{2s_e}(e))$ to $\sigma$ for every $e\in\mathcal E([g]_{\mathcal M})\setminus\mathcal E([\varphi]_{\mathcal M})$ that is not occupied by a dimer, and for every $e\in\left(\mathcal E(g)\setminus\mathcal E(\varphi)\right)\setminus\mathcal E([g]_{\mathcal M})$. +\end{itemize} +Since $\omega_{\varphi}(\chi_{2i}(e))=\omega_{\varphi}(\chi_{2i-1}(e))+1$, the $\omega_{\varphi}(\chi_i(e))$ labels can be ordered and moved after the vertices of $\mathcal V([g]_{\mathcal M})$ by a positive-signature permutation. Furthermore, the vertices of $\mathcal V([g]_{\mathcal M})$ can be ordered by a positive-signature permutation if and only if $\sigma$ is positive. Since $\omega_{\varphi}(\chi_{i}(e))>|g|$, this implies that $\sigma$ is positive if and only if $\lambda_{\varphi,\mathcal M}(\sigma)$ is. +\medskip + +\indent Now note that $\varphi$ is an evenly filled graph, whose direction and labeling is compatible with the recipe given in section \ref{seceven} (see ``\ref{recipeeven}''). +It therefore follows from lemma \ref{lemmaevenpositivity} that +$[\varphi]_{\mathcal M}$ is positive. Moreover, the image of any extended dimer covering of $[g]_{\mathcal M}$ through $\lambda_{\varphi,\mathcal M}$ is positive, which implies that the extended dimer covering itself is positive, and therefore that $[g]_{\mathcal M}$ is as well.\hfill$\square$ +\bigskip + +\indent It follows from lemma~\ref{lemmaenclosedpositivity} and +theorem~\ref{theokasteleyn} that by defining $a(\mathbf d)$ as in~(\ref{eqAdef}), +$\mathrm{pf}([a(\mathbf d)]_{\mathcal M})$ is the partition function of dimers on $[g]_{\mathcal M}$, +which, by theorem~\ref{theolieb}, implies +theorem \ref{theomain} for enclosed graphs, provided $g\in\mathcal G_{en}$ is directed and labeled as above (see ``\ref{recipeenclosed}''). + +\section{Reducing generic planar graphs to enclosed graphs}\label{sec7} +In this section, we discuss how a generic planar graph can be reduced to an enclosed graph. In particular, we will show how to add 0-weight edges to a graph to construct a boundary circuit, and how to add 0-weight edges and vertices to a graph that has an odd number of vertices in its boundary circuit to turn it into an enclosed graph, for which theorem~\ref{theomain} was proved in section~\ref{secenclosed}. + +\subsection{Boundary circuit}\label{secboundarycircuit} +In this section, we give an algorithm to construct a boundary circuit for any planar graph $g$ by adding 0-weight edges. The construction is such that all the vertices of the boundary $\partial g$ of $g$ are in the boundary circuit. The boundary MD partition function on the graph with the boundary circuit is, therefore, equal to that on $g$, and the Pfaffian associated to the graph with the boundary circuit is equal to that associated to $g$. +\bigskip + +\indent First of all, if $g$ is disconnected, then we add 0-weight edges to it to connect its connected components to each other. + +\indent Then, consider a closed path shadowing the boundary of $g$ from the outside, and denote by $(v_1,\cdots,v_n)$ the string (possibly with repetitions) +listing the vertices of the boundary in the order encountered along the path. We identify $v_{n+1}\equiv v_1$. Then consider the ordered sub-set $(v_{i_1},\cdots,v_{i_k})$ of $(v_1,\cdots,v_n)$ obtained by +erasing the repetitions. If a pair $\{v_{i_j},v_{i_{j+1}}\}$, $j=1,\ldots,k$, is not an edge of $g$, then we add a $0$-weight edge from $v_{i_j}$ to $v_{i_{j+1}}$. +By construction, $(v_{i_1},\cdots,v_{i_k})$ is the boundary circuit of the resulting graph. See figure \ref{figboundaryexample} for an example. + +\bigskip + +\begin{figure} +\hfil\includegraphics[width=6cm]{Figs/boundary_circuit_naked.pdf} +\hfil\includegraphics[width=6cm]{Figs/boundary_circuit.pdf}\par\penalty10000 +\medskip +\caption{A graph with no boundary circuit. In the leftmost figure, the edges and vertices of the boundary graph are drawn thicker and {\color{blue}blue}, and the dotted line represents the path shadowing the boundary from the outside, which we think of as starting and ending at $1$. The corresponding string with repetitions is $(1,2,3,4,5,4,6,4,7,4,3,8,9,10,11,12,3,2,13)$. After having erased the repetitions, we obtain the new string $(1,2,3,4,5,6,7,8,9,10,11,12,13)$. All the adjacent pairs but $\{5,6\}$, $\{6,7\}$, $\{7,8\}$ and $\{12,13\}$ are edges of $g$. In the rightmost figure, these four extra edges are added to the graph and drawn thicker and {\color{red}red}.} +\label{figboundaryexample} +\end{figure} + +\indent Once the boundary circuit has been constructed, if the boundary circuit contains an even number of vertices, then the resulting graph is an enclosed graph and can be directed and labeled as in section~\ref{secenclosed}. If the boundary circuit contains an odd number of vertices then the graph can be further reduced to an enclosed graph as explained in the following section. + +\subsection{Making a boundary circuit even}\label{secevenboundarycircuit} +In this section, we show how to add edges and vertices to a graph with a boundary circuit that contains an odd number of vertices to an enclosed graph, in such a way that the boundary MD partition function is unchanged. The procedure is simple: pick an edge of the boundary circuit, and +replace the edge according to figure~\ref{figreplodd}. + +\bigskip + +\begin{figure} +\hfil\parbox[m]{4cm}{\includegraphics[width=4cm]{Figs/oddrepl_bef}} +\hfil$\longmapsto$ +\hfil\parbox[m]{4cm}{\includegraphics[width=4cm]{Figs/oddrepl_aft}}\par\penalty10000 +\medskip +\caption{Replacing an edge of the boundary circuit in order to make it contain an even number of edges. The weight of the thick {\color{blue}blue} (color online) edge is set to 1. Note that since the {\color{blue} blue} edge must be occupied in every boundary MD covering, the weights of the other two extra edges and the weight of the extra vertex on the boundary circuit do not affect the boundary MD partition function.} +\label{figreplodd}\end{figure} + +\indent It is easy to check that, if the extra edge is given weight 1, then the resulting graph has the same boundary MD partition function. In addition, the resulting graph is an enclosed graph, and can be directed and labeled as in section~\ref{secenclosed}, and it is straightforward to check that the Pfaffian computed from the graph with the extra edges and vertices is equal to that without. This concludes the proof of theorem~\ref{theomain}. +\vskip100pt + +\hfil{\Large\bf Acknowledgments}\par +\bigskip +We thank Michael Aizenman and Hugo Duminil-Copin for discussing their work in progress on the random current representation for planar lattice models with us. We gratefully acknowledge financial support from the A*MIDEX project ANR-11-IDEX-0001-02 (A.G.), +from the PRIN National Grant {\it Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions} (A.G. and I.J.), and +NSF grant PHY-1265118 (E.H.L.). +We also would like to thank Tom Spencer and Joel Lebowitz for their hospitality at the IAS in Princeton and their continued interest in this problem. +\vfill +\eject + +\appendix +\section{Examples}\label{app.A} +In this appendix we provide some examples of Pfaffian formulas to illustrate the discussion. + +\subsection{A simple graph} +In this section, we compute the Pfaffian corresponding to the first graph of figure~\ref{figsimpleexample}, directed and labeled as in figure~\ref{figsimpleexamplelabels}. We set the edge weights $d_e=1$ and assume the monomer weights $\ell_i$ are all equal to $\sqrt{x}$. The antisymmetric matrix $A(\bm\ell,\mathbf d)$ is +$$ +A(x)=\left(\begin{array}{cccccccc} +0&1+x&-x&x&-x&x&-x&1+x\\ +&0&1+x&-x&x&-x&1+x&-x\\ +&&0&1+x&1-x&x&1-x&x\\ +&&&0&1+x&-x&x&-x\\ +&&&&0&1+x&-x&x\\ +&&&&&0&1+x&-x\\ +&&&&&&0&1+x\\ +&&&&&&&0 +\end{array}\right) +$$ +which is completed by antisymmetry. The MD partition function is therefore +\begin{equation} +\Xi(x)=\mathrm{pf}(A(x))= +x^4+11\,x^3+33\,x^2+28\,x+3. +\label{eqsimpleexample}\end{equation} + +\subsection{Another simple graph: the L-shape} +In this section we compute the MD partition function for another simple graph: the {\it L-shape}, represented in figure~\ref{figLshape}. We direct and label the graph as per the discussion in section~\ref{secsimple} (see figure~\ref{figLshape}) and find (setting $d_v=1$ and $\ell_i=\sqrt{x}$ as before) +$$ +A(x)=\left(\begin{array}{cccccccc} +0&1+x&-x&x&-x&x&-x&1+x\\ +&0&1+x&-x&1+x&-x&x&-x\\ +&&0&1+x&-x&x&-x&x\\ +&&&0&1+x&-x&x&-x\\ +&&&&0&1+x&-x&1+x\\ +&&&&&0&1+x&-x\\ +&&&&&&0&1+x\\ +&&&&&&&0 +\end{array}\right) +$$ +which is completed by antisymmetry. The MD partition function is therefore +\begin{equation} +\Xi(x)=\mathrm{pf}(A(x))= +x^4+10\,x^3+28\,x^2+24\,x+4 +\label{eqLshape}\end{equation} + +\begin{figure} +\hfil\includegraphics{Figs/Lshape.pdf} +\hfil\includegraphics{Figs/Lshape_label.pdf}\par\penalty10000 +\medskip +\captionshort{The L-shape graph.} +\label{figLshape} +\end{figure} + +\subsection{An evenly filled graph} +In this section, we compute the Pfaffian corresponding to the graph in figure~\ref{figevenexample}, directed and labeled as in figure~\ref{figevenexamplelabels}. We set $d_e=1$ and $\ell_i=\sqrt{x}$. Since the expression of the matrix $A$ is rather long, we split it into lines and only write the $i1% +\seqskip% +\penalty-1000% +\fi% +% hyperref anchor +\hrefanchor% +% define tag (for \label) +\xdef\tag{\sectionprefix\thesectioncount}% +% get widths +\def\@secnum{{\bf\Large\sectionprefix\thesectioncount.\hskip10pt}}% +\settowidth\secnumwidth{\@secnum}% +\setlength\sectitlewidth\textwidth% +\addtolength\sectitlewidth{-\secnumwidth}% +% print name +\parbox{\textwidth}{% +\@secnum% +\parbox[t]{\sectitlewidth}{\Large\bf #1}}% +% write to table of contents +\iftoc% +% save lncount in aux variable which is written to toc +\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@sec.\thesectioncount\endcsname{\thelncount}}% +\write\tocoutput{\noexpand\tocsection{#1}{\thepage}}% +\fi% +\par\penalty10000% +\bigskip\penalty10000% +} + +%% subsection +\def\subsection#1{ +% counters +\stepcounter{subsectioncount}% +\setcounter{pointcount}{0}% +\setcounter{subpointcount}{0}% +\setcounter{subsubpointcount}{0}% +\ifsubseqcount% +\setcounter{seqcount}0% +\fi% +% space before subsection +\subseqskip% +\penalty-500% +% hyperref anchor +\hrefanchor +% define tag (for \label) +\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount}% +% get widths +\def\@secnum{{\bf\large\hskip.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\hskip5pt}}% +\settowidth\secnumwidth{\@secnum}% +\setlength\sectitlewidth\textwidth% +\addtolength\sectitlewidth{-\secnumwidth}% +% print name +\parbox{\textwidth}{% +\@secnum% +\parbox[t]{\sectitlewidth}{\large\bf #1}}% +% write to table of contents +\iftoc% +% save lncount in aux variable which is written to toc +\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsec.\thesectioncount.\thesubsectioncount\endcsname{\thelncount}}% +\write\tocoutput{\noexpand\tocsubsection{#1}{\thepage}}% +\fi% +\par\penalty10000% +\medskip\penalty10000% +} + +%% itemize +\newlength\itemizeskip +% left margin for items +\setlength\itemizeskip{20pt} +% item symbol +\def\itemizept{\textbullet} +\newlength\itemizeseparator +% space between the item symbol and the text +\setlength\itemizeseparator{5pt} +% penalty preceding an itemize +\def\itemizepenalty{0} + +\newlength\current@itemizeskip +\setlength\current@itemizeskip{0pt} +\def\itemize{ + \par\penalty\itemizepenalty\medskip\penalty\itemizepenalty + \addtolength\current@itemizeskip{\itemizeskip} + \leftskip\current@itemizeskip +} +\def\enditemize{ + \addtolength\current@itemizeskip{-\itemizeskip} + \par\leftskip\current@itemizeskip + \medskip +} +\newlength\itempt@total +\def\item{ + \settowidth\itempt@total{\itemizept} + \addtolength\itempt@total{\itemizeseparator} + \par + \medskip + \hskip-\itempt@total\itemizept\hskip\itemizeseparator +} + +%% enumerate +\newcounter{enumerate@count} +\def\enumerate{ + \setcounter{enumerate@count}0 + \let\olditemizept\itemizept% + \def\itemizept{\stepcounter{enumerate@count}\theenumerate@count.} +} +\def\endenumerate{ + \let\itemizept\olditemizept +} + +%% points +\def\point{ + \stepcounter{pointcount} + \setcounter{subpointcount}{0} + % hyperref anchor + \hrefanchor + \indent{\bf \thepointcount\ - } + % define tag (for \label) + \xdef\tag{\thepointcount} +} +\def\subpoint{ + \stepcounter{subpointcount} + \setcounter{subsubpointcount}0 + % hyperref anchor + \hrefanchor + \indent\hskip.5cm{\bf \thepointcount-\thesubpointcount\ - } + % define tag (for \label) + \xdef\tag{\thepointcount-\thesubpointcount} +} +\def\subsubpoint{ + \stepcounter{subsubpointcount} + % hyperref anchor + \hrefanchor + \indent\hskip1cm{\bf \thepointcount-\thesubpointcount-\thesubsubpointcount\ - } + % define tag (for \label) + \xdef\tag{\thepointcount-\thesubpointcount-\thesubsubpointcount} +} +\def\pspoint{ + \stepcounter{pointcount} + \stepcounter{subpointcount} + \setcounter{subsubpointcount}0 + % hyperref anchor + \hrefanchor + \indent\hskip.5cm{\bf \thepointcount-\thesubpointcount\ - } + % define tag (for \label) + \xdef\tag{\thepointcount-\thesubpointcount} +} + +% reset points +\def\resetpointcounter{ + \setcounter{pointcount}{0} + \setcounter{subpointcount}{0} + \setcounter{subsubpointcount}{0} +} + +%% equation numbering +\def\seqcount{ + \stepcounter{seqcount} + % the output + \edef\seqformat{\theseqcount} + % add subsection number + \ifsubseqcount + \let\tmp\seqformat + \edef\seqformat{\thesubsectioncount.\tmp} + \fi + % add section number + \ifsections + \let\tmp\seqformat + \edef\seqformat{\sectionprefix\thesectioncount.\tmp} + \fi + % define tag (for \label) + \xdef\tag{\seqformat} + % write number + \marginnote{\eqnumstyle\hfill(\seqformat)} +} +%% equation environment compatibility +\def\equation{\hrefanchor$$\seqcount} +\def\endequation{$$\@ignoretrue} + +%% figures +\newlength\figwidth +\setlength\figwidth\textwidth +\addtolength\figwidth{-2.5cm} + +\def\caption#1{% +\stepcounter{figcount}% +% hyperref anchor +\hrefanchor% +% the number of the figure +\edef\figformat{\thefigcount}% +% add section number +\ifsections% +\let\tmp\figformat% +\edef\figformat{\sectionprefix\thesectioncount.\tmp}% +\fi% +% define tag (for \label) +\xdef\tag{\figformat}% +% write +\hfil fig \figformat: \parbox[t]{\figwidth}{\leavevmode\small#1}% +\par\bigskip% +} +%% short caption: centered +\def\captionshort#1{% +\stepcounter{figcount}% +% hyperref anchor +\hrefanchor% +% the number of the figure +\edef\figformat{\thefigcount}% +% add section number +\ifsections% +\let\tmp\figformat% +\edef\figformat{\sectionprefix\thesectioncount.\tmp}% +\fi% +% define tag (for \label) +\xdef\tag{\figformat}% +% write +\hfil fig \figformat: {\small#1}% +\par\bigskip% +} + +%% environment +\def\figure{ + \par\penalty-500 +} +\def\endfigure{ + \par\penalty-1000 +} + +%% delimiters +\def\delimtitle#1{\par% +\leavevmode% +\raise.3em\hbox to\hsize{% +\lower0.3em\hbox{\vrule height0.3em}% +\hrulefill% +\ \lower.3em\hbox{#1}\ % +\hrulefill% +\lower0.3em\hbox{\vrule height0.3em}% +}\par\penalty10000} + +%% callable by ref +\def\delimtitleref#1{\par% +% hyperref anchor +\hrefanchor% +% define tag (for \label) +\xdef\tag{#1}% +\leavevmode% +\raise.3em\hbox to\hsize{% +\lower0.3em\hbox{\vrule height0.3em}% +\hrulefill% +\ \lower.3em\hbox{\bf #1}\ % +\hrulefill% +\lower0.3em\hbox{\vrule height0.3em}% +}\par\penalty10000} + +%% no title +\def\delim{\par% +\leavevmode\raise.3em\hbox to\hsize{% +\lower0.3em\hbox{\vrule height0.3em}% +\hrulefill% +\lower0.3em\hbox{\vrule height0.3em}% +}\par\penalty10000} + +%% end delim +\def\enddelim{\par\penalty10000% +\leavevmode% +\raise.3em\hbox to\hsize{% +\vrule height0.3em\hrulefill\vrule height0.3em% +}\par} + +\def\delim{\par\leavevmode\raise.3em\hbox to\hsize{\vrule height0.3em\hrulefill\vrule height0.3em}\par\penalty10000} +\def\enddelim{\par\penalty10000\leavevmode\raise.3em\hbox to\hsize{\vrule height0.3em\hrulefill\vrule height0.3em}\par} + +%% theorem headers +\def\theo#1{ + \stepcounter{Theocount} + % reset points + \ifresetpointattheo\resetpointcounter\fi + % hyperref anchor + \hrefanchor + % the number + \def\formattheo{\theTheocount} + % add section number + \ifsections + \let\tmp\formattheo + \edef\formattheo{\sectionprefix\thesectioncount.\tmp} + \fi + % define tag (for \label) + \xdef\tag{\formattheo} + % write + \delimtitle{\bf #1 \formattheo} +} +\let\endtheo\enddelim +%% theorem headers with name +\def\theoname#1#2{ + \stepcounter{Theocount} + % reset points + \ifresetpointattheo\resetpointcounter\fi + % hyperref anchor + \hrefanchor + % the number + \def\formattheo{\theTheocount} + % add section number + \ifsections + \let\tmp\formattheo + \edef\formattheo{\sectionprefix\thesectioncount.\tmp} + \fi + % define tag (for \label) + \xdef\tag{\formattheo} + % write + \delimtitle{\bf #1 \formattheo\ \rm(\it #2\rm)} +} + +%% start appendices +\def\appendix{% +\vfill +\pagebreak +% counter +\setcounter{sectioncount}0% +% prefix +\def\sectionprefix{A}% +% write +{\bf \LARGE Appendices}\par\penalty10000\bigskip\penalty10000% +% add a mention in the table of contents +\iftoc% +\immediate\write\tocoutput{\noexpand\tocappendices}\penalty10000% +\fi% +%% uncomment for new page for each appendix +%\def\seqskip{\vfill\pagebreak} +} + +%% start references +\def\references{% +\hrefanchor% +% write +{\bf \LARGE References}\par\penalty10000\bigskip\penalty10000% +% add a mention in the table of contents +\iftoc% +% save lncount in aux variable which is written to toc +\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@references\endcsname{\thelncount}}% +\write\tocoutput{\noexpand\tocreferences{\thepage}}\penalty10000% +\fi% +} + + +%% table of contents +\newif\iftocopen +\def\tableofcontents{ +{\bf \large Table of contents:}\par\penalty10000\bigskip\penalty10000% +% copy content from file +\IfFileExists{\jobname.toc}{\input{\jobname.toc}}{{\tt error: table of contents missing}} +% open new toc +\newwrite\tocoutput +\immediate\openout\tocoutput=\jobname.toc +\toctrue +} +%% close file +\AtEndDocument{ + % close toc + \iftoc + \immediate\closeout\tocoutput + \fi +} + + +%% fill line with dots +\def\leaderfill{\leaders\hbox to 1em {\hss. \hss}\hfill} + +%% same as sectionprefix +\def\tocsectionprefix{} + +%% toc formats +\def\tocsection #1#2{ + \stepcounter{tocsectioncount} + \setcounter{tocsubsectioncount}{0} + % write + \smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt {\color{blue}#1}\leaderfill#2}\par +} +\def\tocsubsection #1#2{ + \stepcounter{tocsubsectioncount} + % write + {\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par +} +\def\tocappendices{ + \medskip + \setcounter{tocsectioncount}0 + {\bf Appendices}\par + \smallskip + \def\tocsectionprefix{A} +} +\def\tocreferences#1{ + \medskip + {\hyperlink{ln.\csname toc@references\endcsname}{{\color{blue}\bf References}\leaderfill#1}}\par + \smallskip +} diff --git a/kiss.cls b/kiss.cls new file mode 100644 index 0000000..f610aa4 --- /dev/null +++ b/kiss.cls @@ -0,0 +1,47 @@ +%% +%% Barebones class declaration +%% + +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{kiss} + +\setlength\paperheight {297mm} +\setlength\paperwidth {210mm} + +%% fonts +\input{size11.clo} +\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} +\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} +\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} +\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} +\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} +\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} +\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} + +%% something is wrong with \thepage, redefine it +\gdef\thepage{\the\c@page} + +%% array lines +\setlength\arraycolsep{5\p@} +\setlength\arrayrulewidth{.4\p@} + + +%% default offsets: 1in, correct with \hoffset and \voffset +\hoffset=-50pt +\voffset=-72pt +%% horizontal margins +%\oddsidemargin=31pt +%\evensidemargin=31pt +%% vertical margin +%\topmargin=20pt +%% body size +\textwidth=460pt +\textheight=704pt +%% header size and margin +%\headheight=12pt +%\headsep=25pt +%% footer size +%\footskip=30pt +%% margin size and margin +\marginparwidth=35pt +%\marginparsep=10pt diff --git a/toolbox.sty b/toolbox.sty new file mode 100644 index 0000000..7f5738e --- /dev/null +++ b/toolbox.sty @@ -0,0 +1,46 @@ +%% +%% A collection of useful commands +%% + +%% can call commands even when they are not defined +\def\safe#1{% +\ifdefined#1% +#1% +\else% +{\color{red}\bf?}% +\fi% +} + + +%% larger skip +\newskip\hugeskipamount + \hugeskipamount=24pt plus8pt minus8pt +\def\hugeskip{\vskip\hugeskipamount} + + +%% penalty before large blocks +\def\preblock{ + \penalty-500 +} + +%% listparpenalty prevents page breaks before lists +\newcount\prevparpenalty +\def\listparpenalty{ + \prevparpenalty=\@beginparpenalty + \@beginparpenalty=10000 +} +%% back to previous value +\def\unlistparpenalty{ + \@beginparpenalty=\prevparpenalty +} + +%% stack relations in subscript or superscript +\def\mAthop#1{\displaystyle\mathop{\scriptstyle #1}} + +%% array spanning the entire line +\newlength\largearray@width +\setlength\largearray@width\textwidth +\addtolength\largearray@width{-10pt} +\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]} +\def\endlargearray{\end{array}} + -- cgit v1.2.3-54-g00ecf