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

2011/02/28 by Martin Klazar, Klazar, Martin · 1 citation
Materials Science · Mathematics · #05A16 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Quasicrystal Structures and Properties #Stochastic processes and statistical mechanics #math.CO #math.PR #msc:05A16

paper · pdf · doi:10.48550/arxiv.1102.5733

16 pages. Minor corrections of typos, errors and formulations

openalex publication_date 2011/02/28 · arxiv created 2011/04/07 · arxiv updated 2011/04/08 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

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))²x² + ... .

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