84cec16dd8
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.
32 lines
1.0 KiB
TeX
32 lines
1.0 KiB
TeX
\documentclass{standalone}
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\usepackage{tikz}
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\usepackage{dimer}
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\begin{document}
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\begin{tikzpicture}
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%% mantles
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\fill[color=gray](2,1.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
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\fill[color=white](4,2.5)--++(9,0)--++(0.5,0.5)--++(0,6)--++(-0.5,0.5)--++(-9,0)--++(-0.5,-0.5)--++(0,-6)--cycle;
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\fill[color=gray](4,3.5)--++(4,0)--++(0.5,0.5)--++(0,4)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-4)--cycle;
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\fill[color=white](5,5.5)--++(2,0)--++(0.5,0.5)--++(0,0)--++(-0.5,0.5)--++(-2,0)--++(-0.5,-0.5)--++(0,-0)--cycle;
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%% segments
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\foreach \i in {4,...,8}{
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\draw[color=gray, line width=15pt](8.5,\i)--++(5,0);
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}
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%% grid
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\grid{17}{12}{[color=lightgray](0,0)}
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%% loops
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\draw[color=black, line width=5pt](2,1.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
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\draw[color=black, line width=5pt](3.5,4)--++(0.5,-0.5)--++(4,0)--++(0.5,0.5)--++(0,4)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5);
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\draw[color=red, line width=10pt](3.5,8)--++(0,-4);
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\end{tikzpicture}
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\end{document}
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