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

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Initial commit
2017-09-13 23:29:54 +00:00
\documentclass{standalone}
\usepackage{tikz}
\usepackage{dimer}
\begin{document}
\begin{tikzpicture}
%% mantles
% blue
\fill[color=blue](2,6.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
\fill[color=white](4,7.5)--++(9,0)--++(0.5,0.5)--++(0,6)--++(-0.5,0.5)--++(-9,0)--++(-0.5,-0.5)--++(0,-6)--cycle;
% red
\fill[color=red](6,8.5)--++(4,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
\fill[color=white](7,10.5)--++(2,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-2,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
% teal
\fill[color=teal](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;
\fill[color=white](11,20.5)--++(7,0)--++(0.5,-0.5)--++(0.5,-0.5)--++(1,0)--++(0.5,-0.5)--++(0,-9)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,20)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-9)--cycle;
% magenta
\fill[color=magenta](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
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
\fill[color=cyan](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
\fill[color=white](26,13.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
Initial commit
2017-09-13 23:29:54 +00:00
% orange
\fill[color=orange](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;
% green
\fill[color=green](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
%% segments
\foreach \i in {0,...,1}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,37);
}
\foreach \i in {2,...,8}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,7);
\draw[color=gray, line width=15pt](\i,15.5)--++(0,21);
}
\foreach \i in {9,...,15}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,7);
\draw[color=gray, line width=15pt](\i,15.5)--++(0,4);
\draw[color=gray, line width=15pt](\i,31.5)--++(0,5);
}
\foreach \i in {16,...,18}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,20);
\draw[color=gray, line width=15pt](\i,31.5)--++(0,5);
}
\foreach \i in {19,...,34}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,9);
\draw[color=gray, line width=15pt](\i,31.5)--++(0,5);
}
\foreach \i in {35,...,36}{
\draw[color=gray, line width=15pt](\i,-0.5)--++(0,37);
}
\draw[color=blue, line width=15pt](3.5,8)--++(10,0);
\foreach \i in {9,...,14}{
\draw[color=blue, line width=15pt](3.5,\i)--++(2,0);
\draw[color=blue, line width=15pt](10.5,\i)--++(3,0);
}
\foreach \i in {7,...,9}{
\draw[color=red, line width=15pt](\i,10.5)--++(0,2);
}
\foreach \i in {10,...,11}{
\draw[color=teal, line width=15pt](20.5,\i)--++(12,0);
}
\foreach \i in {12,...,17}{
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
\draw[color=teal, line width=15pt](20.5,\i)--++(4,0);
\draw[color=teal, line width=15pt](28.5,\i)--++(4,0);
Initial commit
2017-09-13 23:29:54 +00:00
}
\foreach \i in {18,...,19}{
\draw[color=teal, line width=15pt](20.5,\i)--++(12,0);
}
\draw[color=teal, line width=15pt](18.5,20)--++(14,0);
\draw[color=teal, line width=15pt](10.5,21)--++(22,0);
\foreach \i in {22,...,29}{
\draw[color=teal, line width=15pt](10.5,\i)--++(2,0);
\draw[color=teal, line width=15pt](31.5,\i)--++(1,0);
}
\draw[color=teal, line width=15pt](10.5,30)--++(22,0);
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
\foreach \i in {26,...,27}{
Initial commit
2017-09-13 23:29:54 +00:00
\draw[color=cyan, line width=15pt](\i,13.5)--++(0,2);
}
\draw[color=magenta, line width=15pt](14,23.5)--++(0,4);
\foreach \i in {21,...,22}{
\draw[color=magenta, line width=15pt](\i,23.5)--++(0,4);
}
\foreach \i in {23,...,26}{
\draw[color=magenta, line width=15pt](\i,23.5)--++(0,1);
\draw[color=magenta, line width=15pt](\i,26.5)--++(0,1);
}
\foreach \i in {27,...,30}{
\draw[color=magenta, line width=15pt](\i,23.5)--++(0,4);
}
\foreach \i in {25,...,26}{
\draw[color=orange, line width=15pt](16.5,\i)--++(2,0);
}
%% grid
\grid{36}{36}{[color=lightgray](0,0)}
%% loops
% blue
\draw[color=black, line width=5pt](2,6.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
% red
\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;
% teal
\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;
% magenta
\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;
% cyan
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
\draw[color=black, line width=5pt](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
Initial commit
2017-09-13 23:29:54 +00:00
% orange
\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;
%green
\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;
\end{tikzpicture}
\end{document}
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