From 84cec16dd8bbab11a43f00380a1e6354e3aa0c04 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Wed, 25 Apr 2018 22:10:43 +0000 Subject: Update to v0.1: Changed: Slight change in the statement of the main theorem: no long-range positional order between any dimers, regardless of orientation. Changed: Section 4 was completely reworked. It now consists of a discussion, in which definitions and figures appear as needed. The main lemma of the section was moved to the end. Fixed: The definition of external contours had a serious typo in it. Fixed: The definition of non-trivial polymers was backwards. What were called non-trivial polymers are actually trivial, and vice versa. Changed: The contour and polymer figures were changed to include an example of a polymer with a non-trivial recursive structure. Fixed: The clusters appearing in cluster expansions must be defined using multisets instead of sets, as they were before. The notation was adjusted accordingly. Fixed: The expression of the Ursell function was missing a combinatorial factor. Fixed: When splitting a sum over clusters into a sum over a polymer and the rest of the cluster, the sum over the multiplictity of the polymer must be separated as well, in order to avoid overcounting. Fixed: In the proof of lemma 5.2, the bound on \mathfrak x required a better estimate on b_+ and nu_+. Fixed: In the proof of lemma 5.2, the argument for why the number of bad edges on the boundary of \iota is bounded by \ell_0(l-m) was wrong. Added: In the proof of lemma 5.2, a figure was added to illustrate the issue with bad edges. Added: Some more minor points have been expanded to include more details. Fixed: Miscellaneous typos and reformating. --- figs/polymer_example.fig/lollipops.tikz.tex | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) (limited to 'figs/polymer_example.fig/lollipops.tikz.tex') diff --git a/figs/polymer_example.fig/lollipops.tikz.tex b/figs/polymer_example.fig/lollipops.tikz.tex index 6b6a058..e1a9df1 100644 --- a/figs/polymer_example.fig/lollipops.tikz.tex +++ b/figs/polymer_example.fig/lollipops.tikz.tex @@ -21,8 +21,7 @@ \fill[color=teal](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle; \fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle; % cyan -\fill[color=cyan](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle; -\fill[color=white](24,13.5)--++(5,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-1)--cycle; +\fill[color=cyan](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle; % orange \fill[color=teal](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle; \fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle; @@ -36,8 +35,6 @@ \draw[color=red, line width=3pt](3.5,12)--++(2,0); -\draw[color=cyan, line width=3pt](20.5,14)--++(2,0); - \foreach \i in {22,...,29}{ \draw[color=teal, line width=3pt](31.5,\i)--++(1,0); } @@ -47,6 +44,8 @@ \draw[color=teal, line width=3pt](\i,26.5)--++(0,1); } +\draw[color=cyan, line width=3pt](25,13.5)--++(0,-4); + %% grid \grid{36}{36}{[color=lightgray](0,0)} @@ -64,8 +63,8 @@ \draw[color=black, line width=1pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle; -\draw[color=black, line width=1pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle; -\draw[color=black](23,18.5)node{\it e}; +\draw[color=black, line width=1pt](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle; +\draw[color=black](23.5,18)node{\it e}; \draw[color=black, line width=1pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle; -- cgit v1.2.3-70-g09d2