diff options
author | Ian Jauslin <jauslin@ias.edu> | 2018-04-25 22:10:43 +0000 |
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committer | Ian Jauslin <jauslin@ias.edu> | 2018-04-26 05:27:22 +0000 |
commit | 84cec16dd8bbab11a43f00380a1e6354e3aa0c04 (patch) | |
tree | 3346280faa020c91cdebe9088082eaaf48165009 /figs/polymer_example.fig/polymers.tikz.tex | |
parent | 52a5942994007acc29107f036c9b8f386b19399f (diff) |
Update to v0.1: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.
Diffstat (limited to 'figs/polymer_example.fig/polymers.tikz.tex')
-rw-r--r-- | figs/polymer_example.fig/polymers.tikz.tex | 76 |
1 files changed, 0 insertions, 76 deletions
diff --git a/figs/polymer_example.fig/polymers.tikz.tex b/figs/polymer_example.fig/polymers.tikz.tex deleted file mode 100644 index e2f688a..0000000 --- a/figs/polymer_example.fig/polymers.tikz.tex +++ /dev/null @@ -1,76 +0,0 @@ -\documentclass{standalone} - -\usepackage{tikz} -\usepackage{dimer} - -\begin{document} -\begin{tikzpicture} - -%% mantles - -% blue -\fill[color=gray](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=gray](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=gray](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=gray](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=gray](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=gray](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=gray](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 - -\draw[color=gray, line width=15pt](4,-0.5)--++(0,7); -\draw[color=gray, line width=15pt](10,15.5)--++(0,4); -\draw[color=gray, line width=15pt](11,15.5)--++(0,4); -\draw[color=gray, line width=15pt](9,31.5)--++(0,5); - -\foreach \i in {9,...,14}{ - \draw[color=gray, line width=15pt](3.5,\i)--++(2,0); - \draw[color=gray, line width=15pt](10.5,\i)--++(3,0); -} - -\foreach \i in {12,...,17}{ - \draw[color=gray, line width=15pt](20.5,\i)--++(2,0); - \draw[color=gray, line width=15pt](30.5,\i)--++(2,0); -} -\foreach \i in {22,...,29}{ - \draw[color=gray, line width=15pt](10.5,\i)--++(2,0); - \draw[color=gray, line width=15pt](31.5,\i)--++(1,0); -} - -\foreach \i in {23,...,26}{ - \draw[color=gray, line width=15pt](\i,23.5)--++(0,1); - \draw[color=gray, line width=15pt](\i,26.5)--++(0,1); -} - -%% grid -\grid{36}{36}{[color=lightgray](0,0)} - - -%% loops - -\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; -\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; - -\end{tikzpicture} -\end{document} - |