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17jlC/figs/polymer_example.fig/iota_boundary.tikz.tex

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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.
2018-04-25 22:10:43 +00:00
\documentclass{standalone}
\usepackage{tikz}
\usepackage{dimer}
\begin{document}
\begin{tikzpicture}
%% mantles
\fill[color=gray](2,10.5)--++(18,0)--++(0.5,-0.5)--++(0,-4)--++(-0.5,-0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-3)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,26)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-17)--cycle;
\fill[color=white](4,11.5)--++(17,0)--++(1,-1)--++(0.5,-0.5)--++(0,-4)--++(-1.5,-1.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--++(0.5,-0.5)--++(7,0)--++(0.5,0.5)--++(0,24)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-15)--cycle;
\fill[color=gray](7,16.5)--++(10,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-10,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
\fill[color=white](8,18.5)--++(8,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-8,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
% iota
\fill[color=cyan](4,11.5)--++(17,0)--++(1.5,-1.5)--++(3,0)--++(0,17)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-3)--++(3,0)--++(0.5,0.5)--++(10,0)--++(0.5,-0.5)--++(0,-7)--++(-0.5,-0.5)--++(-10,0)--++(-0.5,.5)--++(-3,0)--++(0,-5)--cycle;
%% segments
\draw[color=red, line width=20pt](3.5,17)--++(3,0);
\draw[color=red, line width=20pt](3.5,24)--++(3,0);
\foreach \i in {18,...,23}{
\draw[color=gray, line width=15pt](3.5,\i)--++(3,0);
}
\draw[color=red, line width=20pt](22.5,10)--++(3,0);
\foreach \i in {6,...,9}{
\draw[color=gray, line width=15pt](22.5,\i)--++(3,0);
}
%% grid
\grid{29}{30}{[color=lightgray](0,0)}
%% loops
\draw[color=black, line width=5pt](2,10.5)--++(18,0)--++(0.5,-0.5)--++(0,-4)--++(-0.5,-0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-3)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,26)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-17)--cycle;
\draw[color=black, line width=5pt](7,16.5)--++(10,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-10,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
\end{tikzpicture}
\end{document}