In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding 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 self-avoiding walks). We modify this model by restricting to a half-plane 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 self-avoiding 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 self-avoiding bridges of height T, taken at its critical point 1/ μ, tends to 0 as T increases, as predicted from SLE theory.
Communications in Mathematical Physics
- Pub Date:
- March 2014
- Mathematical Physics;
- Condensed Matter - Statistical Mechanics;
- Mathematics - Combinatorics
- Major revision, references updated, 25 pages, 13 figures