From 84cec16dd8bbab11a43f00380a1e6354e3aa0c04 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Wed, 25 Apr 2018 22:10:43 +0000 Subject: Update to 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. --- figs/dimer_example.fig/Makefile | 2 +- figs/dimer_example.fig/dimer_conf.py | 8 ++++++++ 2 files changed, 9 insertions(+), 1 deletion(-) (limited to 'figs/dimer_example.fig') diff --git a/figs/dimer_example.fig/Makefile b/figs/dimer_example.fig/Makefile index 8ed4891..b5e148e 100644 --- a/figs/dimer_example.fig/Makefile +++ b/figs/dimer_example.fig/Makefile @@ -1,4 +1,4 @@ -PROJECTNAME=interaction boundary bounding_loop dimer_contour +PROJECTNAME=interaction boundary bounding_loop dimer_loops dimer_contour LIBS=$(notdir $(wildcard libs/*)) PDFS=$(addsuffix .pdf, $(PROJECTNAME)) 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) -- cgit v1.2.3-70-g09d2