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, 8 insertions, 0 deletions

diff --git a/figs/dimer_example.fig/dimer_conf.py b/figs/dimer_example.fig/dimer_conf.py
index 7cff9f1..a570b91 100644
--- a/figs/dimer_example.fig/dimer_conf.py
+++ b/figs/dimer_example.fig/dimer_conf.py
@@ -274,6 +274,14 @@ print("\\grid{"+str(4*l0+3)+"}{"+str(13)+"}{("+str(mu-2*l0-2)+","+str(mu-7)+")}\
 draw_loops([loop1],bounding_loop,"black")
 close_tikz(bounding_loop)
 
+loops=init_tikz("dimer_loops.tikz.tex")
+print("\\grid{"+str(L-1)+"}{"+str(L-1)+"}{(0,0)}\n", file=loops)
+draw_loops([loop1,loop2,loop3],loops,"blue")
+draw_interactions(v_interactions,h_interactions,loops)
+draw_dimers(v_dimers,h_dimers,loops,"black")
+draw_dimers(v_mantle,h_mantle,loops,"gray")
+close_tikz(loops)
+
 contour=init_tikz("dimer_contour.tikz.tex")
 print("\\draw[color=green, line width=15pt]("+str(mu-2-0.5)+","+str(mu+1)+")--++(1,0);", file=contour)
 print("\\draw[color=green, line width=15pt]("+str(mu-2-0.5)+","+str(mu+2)+")--++(1,0);", file=contour)