# The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$

Abstract : 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 $\mu=\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with $n\in [-2,2]$ (the case $n=0$ corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a fugacity associated with boundary sites (also called \emph{surface} sites) and obtain a generalisation of Smirnov's identity. The value of the \emph{critical} surface fugacity was conjectured by Batchelor and Yung in 1995. This value also plays a crucial role in our identity, which thus provides an independent prediction for it. For the case $n=0$, corresponding to SAWs interacting with a surface, we prove the conjectured value of the critical surface fugacity. A critical part of this proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$ increases, as predicted from SLE theory.
Document type :
Journal articles
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https://hal.archives-ouvertes.fr/hal-00719647
Contributor : Mireille Bousquet-Mélou Connect in order to contact the contributor
Submitted on : Friday, July 20, 2012 - 2:09:54 PM
Last modification on : Monday, December 20, 2021 - 4:50:10 PM

### Identifiers

• HAL Id : hal-00719647, version 1
• ARXIV : 1109.0358

### Citation

Nicholas R. Beaton, Mireille Bousquet-Mélou, Jan de Gier, Hugo Duminil-Copin, Anthony J. Guttmann. The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$. Communications in Mathematical Physics, Springer Verlag, 2014, 326, pp.727-754. ⟨hal-00719647⟩

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