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The critical fugacity for surface adsorption of selfavoiding walks on the honeycomb lattice is 1 +√2 

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Submitted to  Communications in Mathematical Physics. 2013  
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 $mu=sqrt{2+sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with $nin [2,2]$ (the case $n=0$ corresponding to SAWs). 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 $y_{ m c}=1+2/sqrt{2n}.$ This value plays a crucial role in our generalized identity, just as the value of growth constant did in Smirnov's identity. For the case $n=0$, corresponding to saws 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 $1/mu$, tends to 0 as $T$ increases, as predicted from SLE theory.  
Identifiers  arXiv: 1109.0358  
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Citation (ISO format)  BEATON, Nicholas R. et al. The critical fugacity for surface adsorption of selfavoiding walks on the honeycomb lattice is 1 +√2. Submitted to: Communications in Mathematical Physics, 2013. https://archiveouverte.unige.ch/unige:30894 