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Update to v0.1:

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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.
This commit is contained in:
Ian Jauslin
2018-04-25 22:10:43 +00:00
parent 52a5942994
commit 84cec16dd8
13 changed files with 1105 additions and 904 deletions
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@@ -0,0 +1,40 @@
0.1
* 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.