arXiv subjects

Nicholas R. Beaton

Publications and source records attributed to Nicholas R. Beaton.

22 records · Page 2Linked to original sources

arXiv2014

Adsorption of self-avoiding walks at a defect

We consider the model of self-avoiding walks on the $d$-dimensional hypercubic lattice interacting with a $d^*$-dimensional defect, where $1\leq d^*<d$. Such an interaction can be attractive or repulsive, and is controlled by a Boltzmann weight $a$ associated with visits to the defect. When $d=3$ and $d^*=1$ or $2$, this can be seen as a model of long linear polymers in a good solvent, interacting with a linear filament or the interface of two liquids of different density. For all combinations of dimensions, there is a critical value $a_{\rm c}$ which separates the desorbed and adsorbed phases of the model. We prove that in all cases $a_{\rm c}=1$, confirming conjectures by a number of authors.

cond-mat.stat-mech↗

arXiv2013

The critical surface fugacity of self-avoiding walks on a rotated honeycomb lattice

In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Their proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. We consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the lattice rotated by $π/2$. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Batchelor, Bennett-Wood and Owczarek. Using similar methods to Beaton et al, we prove that this is indeed the critical fugacity.

math-ph↗

arXiv2013

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

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 $μ=\sqrt{2+\sqrt{2}}.$ 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 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 $y_{\rm c}=1+2/\sqrt{2-n}.$ 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 self-avoiding bridges of height $T$, taken at its critical point $1/μ$, tends to 0 as $T$ increases, as predicted from SLE theory.

math-ph↗

arXiv2011

Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also \emph{prove} the corresponding result fo the edge-weighted case for the honeycomb lattice.

math-ph↗