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/contours.tikz.tex | 8 -------- 1 file changed, 8 deletions(-) (limited to 'figs/polymer_example.fig/contours.tikz.tex') diff --git a/figs/polymer_example.fig/contours.tikz.tex b/figs/polymer_example.fig/contours.tikz.tex index c49c0cc..995bda2 100644 --- a/figs/polymer_example.fig/contours.tikz.tex +++ b/figs/polymer_example.fig/contours.tikz.tex @@ -20,9 +20,6 @@ % magenta \fill[color=blue](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=blue](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; % orange \fill[color=blue](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; @@ -37,10 +34,6 @@ \draw[color=red, line width=15pt](10.5,\i)--++(3,0); } -\foreach \i in {12,...,17}{ - \draw[color=blue, line width=15pt](20.5,\i)--++(2,0); - \draw[color=blue, line width=15pt](30.5,\i)--++(2,0); -} \foreach \i in {22,...,29}{ \draw[color=blue, line width=15pt](10.5,\i)--++(2,0); \draw[color=blue, line width=15pt](31.5,\i)--++(1,0); @@ -60,7 +53,6 @@ \draw[color=black, line width=5pt](6,8.5)--++(4,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-5)--cycle; \draw[color=black, line width=5pt](9,19.5)--++(9,0)--++(0.5,-0.5)--++(0,-10)--++(0.5,-0.5)--++(15,0)--++(0.5,0.5)--++(0,22)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-11)--cycle; \draw[color=black, line width=5pt](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=5pt](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, line width=5pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle; \draw[color=black, line width=5pt](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle; -- cgit v1.2.3-70-g09d2