Compressed self-avoiding walks, bridges and polygons

Abstract

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.

1324-avoiding permutations revisited

“…where estimates of the parameters were given. In the present paper, we present a new, substantially improved algorithm that allows us to give 14 further terms 1 .…”

“…He went on to predict the value of the exponent g in that case. For self-avoiding walks and polygons attached to a surface and pushed toward the surface by a force applied at their top vertex, Beaton et al [1] gave probabilistic arguments for stretched exponential behaviour, but with growth µ n 3/7 1 . Such stretched exponential behaviour is also seen in other combinatorial problems.…”

Self-avoiding walks adsorbed at a surface and pulled at their mid-point

“…The situation can be adapted to model the adsorption of linear polymers at an impenetrable surface [5,10,15,23] and the general features of the adsorption behaviour are now quite well understood. With the invention of micro-manipulation techniques such as atomic force microscopy (AFM) and optical tweezers that allow individual polymer molecules to be pulled [7,26] there has been renewed interest in how polymers respond to a force and, specifically, how self-avoiding walk models of polymers respond to a force [1,2,8,9,14,17]. There has also been some work on how lattice polygons (a model of ring polymers) respond to a force [2,12,13].…”

“…Consequently it is natural to ask how the behaviour depends on where on the polymer the force is being applied. Apart from the case discussed above where the force is applied at the last monomer the only situation that has been studied [2,17,18] is as follows. Suppose that we imagine a plane, parallel to the adsorbing plane, containing the monomers that are furthest away from the adsorbing plane, and apply the force either to pull this plane away from or push it towards the adsorbing surface.…”

Polygons pulled from an adsorbing surface

“…However if the polygon sits at a surface and a compressive force (i.e. y < 1) is applied to the top of the polygon, then it has recently been shown by Beaton et al [4] from probability arguments and particularly assuming SLE predictions that the expected asymptotics now includes a stretched-exponential term. More precisely, for the square lattice,…”