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.

1 files changed, 31 insertions, 0 deletions

diff --git a/figs/polymer_example.fig/mbbX.tikz.tex b/figs/polymer_example.fig/mbbX.tikz.tex
new file mode 100644
index 0000000..61db917
--- /dev/null
+++ b/figs/polymer_example.fig/mbbX.tikz.tex
@@ -0,0 +1,31 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usepackage{dimer}
+
+\begin{document}
+\begin{tikzpicture}
+
+%% mantles
+\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;
+\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;
+\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;
+\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;
+
+%% segments
+\foreach \i in {4,...,8}{
+  \draw[color=gray, line width=15pt](8.5,\i)--++(5,0);
+}
+
+%% grid
+\grid{17}{12}{[color=lightgray](0,0)}
+
+
+%% loops
+\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;
+\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);
+\draw[color=red, line width=10pt](3.5,8)--++(0,-4);
+
+\end{tikzpicture}
+\end{document}
+