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, 38 insertions, 0 deletions

diff --git a/figs/polymer_example.fig/iota_boundary.tikz.tex b/figs/polymer_example.fig/iota_boundary.tikz.tex
new file mode 100644
index 0000000..af2fa37
--- /dev/null
+++ b/figs/polymer_example.fig/iota_boundary.tikz.tex
@@ -0,0 +1,38 @@
+\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}
+