The Critical Fugacity for Surface Adsorption of SelfAvoiding Walks on the Honeycomb Lattice is
Abstract
In 2010, DuminilCopin and Smirnov proved a longstanding conjecture of Nienhuis, made in 1982, that the growth constant of selfavoiding walks on the hexagonal (a.k.a. honeycomb) lattice is . A key identity used in that proof was later generalised by Smirnov so as to apply to a general O( n) loop model with (the case n = 0 corresponding to selfavoiding walks). We modify this model by restricting to a halfplane and introducing a surface fugacity y associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov's identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be . This value plays a crucial role in our generalized identity, just as the value of the growth constant did in Smirnov's identity. For the case n = 0, corresponding to selfavoiding walks interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of selfavoiding bridges of height T, taken at its critical point 1/ μ, tends to 0 as T increases, as predicted from SLE theory.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 March 2014
 DOI:
 10.1007/s0022001418961
 arXiv:
 arXiv:1109.0358
 Bibcode:
 2014CMaPh.326..727B
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Combinatorics
 EPrint:
 Major revision, references updated, 25 pages, 13 figures