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, 5 insertions, 6 deletions

diff --git a/figs/polymer_example.fig/lollipops.tikz.tex b/figs/polymer_example.fig/lollipops.tikz.tex
index 6b6a058..e1a9df1 100644
--- a/figs/polymer_example.fig/lollipops.tikz.tex
+++ b/figs/polymer_example.fig/lollipops.tikz.tex
@@ -21,8 +21,7 @@
 \fill[color=teal](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=cyan](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;
+\fill[color=cyan](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % orange
 \fill[color=teal](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;
@@ -36,8 +35,6 @@
 
 \draw[color=red, line width=3pt](3.5,12)--++(2,0);
 
-\draw[color=cyan, line width=3pt](20.5,14)--++(2,0);
-
 \foreach \i in {22,...,29}{
   \draw[color=teal, line width=3pt](31.5,\i)--++(1,0);
 }
@@ -47,6 +44,8 @@
   \draw[color=teal, line width=3pt](\i,26.5)--++(0,1);
 }
 
+\draw[color=cyan, line width=3pt](25,13.5)--++(0,-4);
+
 %% grid
 \grid{36}{36}{[color=lightgray](0,0)}
 
@@ -64,8 +63,8 @@
 
 \draw[color=black, line width=1pt](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=1pt](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](23,18.5)node{\it e};
+\draw[color=black, line width=1pt](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+\draw[color=black](23.5,18)node{\it e};
 
 \draw[color=black, line width=1pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;