Nicholas R. Beaton

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 (SAW) interacting with (alternate)sitesonthesurfaceofthehoneycomblatticeis1+ √ 2.Akeyidentity used in that proof depends on the existence of a parafermionic observable for SAW interacting with a surface on the honeycomb lattice. Despite the absence ofacorrespondingobservableforSAWonthesquareandtriangularlattices,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 prove the corresponding result for the edge-weighted case for the honeycomb lattice.

Compressed self-avoiding walks, bridges and polygons*

We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.

A numerical adaptation of self-avoiding walk identities from the honeycomb to other 2D lattices

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis that the connective constant of self-avoiding walks (SAWs) on the honeycomb lattice is

The critical pulling force for self-avoiding walks

Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight

Some New Self-avoiding Walk and Polygon Models

y = e^f

The unusual asymptotics of three-sided prudent polygons

associated with the pulling force. This model is known to have a critical point at a certain value

Two-sided prudent walks: a solvable non-directed model of polymer adsorption

y_c

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

of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value

The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is {1+\sqrt{2}}

y_c=1

The enumeration of prudent polygons by area and its unusual asymptotics

has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of this result, and show that further properties of the free energy of this system can be determined by re-interpreting existing results about the two-point function of self-avoiding walks.