Journal article
The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is 1 √2
NR Beaton, M Bousquet-Mélou, J de Gier, H Duminil-Copin, AJ Guttmann
Communications in Mathematical Physics | Published : 2014
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 μ = √2+√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 ∈ [-2,2] (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 yc = 1+2/√2-n. This v..
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Awarded by Australian Research Council
Funding Acknowledgements
We thank Neal Madras, Andrew Rechnitzer, Stu Whittington and Alain Yger for helpful conversations. AJG and JdG acknowledge financial support from the Australian Research Council. NRB was supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS). HDC was supported by the ANR grant BLAN06-3-134462, the ERC AG CONFRA, as well as by the Swiss FNS. Part of this work was carried out during the authors' visits to the Mathematical Sciences Research Institute in Berkeley, during the Spring 2012 Random Spatial Processes Program. The authors thank the Institute for its hospitality and the NSF (grant DMS-0932078) for its financial support.