The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is \(1+\sqrt{2}\). (English) Zbl 1288.82006
Summary: In 2010, H. Duminil-Copin and S. Smirnov [Ann. Math. (2) 175, No. 3, 1653–1665 (2012; Zbl 1253.82012)] 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 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 \(y_{\mathrm c}=1+2/\sqrt{2-n}\). 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 \(1/\mu\), tends to 0 as \(T\) increases, as predicted from SLE theory.
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 \(y_{\mathrm c}=1+2/\sqrt{2-n}\). 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 \(1/\mu\), tends to 0 as \(T\) increases, as predicted from SLE theory.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 82B27 | Critical phenomena in equilibrium statistical mechanics |
Citations:
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