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 y = e f associated with the pulling force. This model is known to have a critical point at a certain value y c of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value y c = 1 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.

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 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 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/µ, tends to 0 as T increases, as predicted from SLE theory.

We study various self-avoiding walks (SAWs) which are constrained to lie in the upper halfplane 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.

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 + √ 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 prove the corresponding result for the edge-weighted case for the honeycomb lattice.

We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z 3 . These polygons are subjected to a force f , parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits f → ±∞. We conjecture that the f → −∞ asymptote is the same as the limiting free energy of "Hamiltonian" polygons, polygons which visit every vertex in a L × M × N box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes.Dedicated to Anthony J. Guttmann on the occasion of his 70 th birthday. arXiv:1604.07465v1 [math-ph]