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/loops.tikz.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'figs/polymer_example.fig/loops.tikz.tex') diff --git a/figs/polymer_example.fig/loops.tikz.tex b/figs/polymer_example.fig/loops.tikz.tex index db06058..a25747f 100644 --- a/figs/polymer_example.fig/loops.tikz.tex +++ b/figs/polymer_example.fig/loops.tikz.tex @@ -20,7 +20,7 @@ \draw[color=magenta, 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](13,30.5)node{\it d}; -\draw[color=cyan, 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=cyan, line width=1pt](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle; \draw[color=black](23,18.5)node{\it e}; \draw[color=orange, 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