On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice

Details On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice

2011-02-28

arxiv.org/abs/1102.5733v2

Mathematics

Combinatorics

16 pages. Minor corrections of typos, errors and formulations

Scientific paper

This is an exposition of the theorem from the title, which says that the number of self-avoiding walks with n steps in the hexagonal lattice has asymptotics (2cos(pi/8))^{n+o(n)}. We lift the key identity to formal level and simplify the part of the proof bounding the growth constant from below. In our calculation the lower bound comes from an identity asserting that a linear combination of 288 generating functions counting self-avoiding walks in a certain domain by length, final edge direction and winding number modulo 48 equals the geometric series 2cos(pi/8)x + (2cos(pi/8))^2x^2 + ... .

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