Investigating the relationship between restriction measures and self-avoiding walks

It is widely believed that the scaling limit of the self-avoiding walk (SAW) is given by Schramm's SLE8/3. In fact, it is known that if SAW has a scaling limit which is conformally invariant, then the distribution of such a scaling limit must be given by SLE 8/3. The purpose of this paper is to study the relationship between SAW and SLE8/3, mainly through the use of restriction measures; conformally invariant measures that satisfy a certain restriction property.;Restriction measures are stochastic processes on randomly growing fractal subsets of the complex plane called restriction hulls, though it turns out that SLE8/3 measure is also a restriction measure. Since SAW should converge to SLE8/3 in the scaling limit, it is thought that many important properties of SAW might also hold for restriction measures, or at the very least, for SLE 8/3.;In Dyhr, 2011, it was shown that if one conditions an infinite length self-avoiding walk in half-plane to have a bridge height at y – 1, and then considers the walk up to height y, then one obtains the distribution of self-avoiding walk in the strip of height y. We show in this paper that a similar result holds for restriction measures Pa , with α ∈ [5/8, 1). That is, if one conditions a restriction hull to have a bridge point at some z ∈ H , and considers the hull up until the time it reaches z, then the resulting hull is distributed according to a restriction measure in the strip of height Im(z). This relies on the fact that restriction hulls contain bridge points a.s. for α ∈ [5/8, 1), which was shown in Alberts, 2010.;We then proceed to show that a more general form of that result holds for restriction hulls of the same range of parameters α. That is, if one conditions on the event that a restriction hull in H passes through a smooth curve γ at a single point, and then considers the hull up to the time that it reaches the point, then the resulting hull is distributed according to a restriction hull in the domain which lies underneath the curve γ. We then show that a similar result holds in simply connected domains other than H .;Next, we conjecture the existence of an object called the infinite length quarter-plane self-avoiding walk. This is a measure on infinite length self-avoiding walks, restricted to lie in the quarter plane. In fact, what we show is that the existence of such a measure depends only on the validity of a relation similar to Kesten's relation for irreducible bridges in the half-plane. The corresponding equation for irreducible bridges in the quarter plane, Conjecture 4.1.19, is believed to be true, and given this result, we show that a measure on infinite length quarter-plane self-avoiding walks analogous to the measure on infinite length half-plane self-avoiding walks (which was proven to exist in LSW, 2002) exists. We first show that, given Conjecture 4.1.19, the measure can be constructed through a concatenation of a sequence of irreducible quarter-plane bridges, and then we show that the distributional limit of the uniform measure on finite length quarter-plane SAWs exists, and agrees with the measure which we have constructed. It then follows as a consequence of the existence of such a measure, that quarter-plane bridges exist with probability 1.;As a follow up to the existence of the measure on infinite length quarter-plane SAWs, and the a.s. existence of quarter-plane bridge points, we then show that quarter plane bridge points exist for restriction hulls of parameter α ∈ [5/8, 3/4), and we calculate the Hausdorff measure of the set of all such bridge points.;Finally, we introduce a new type of (conjectured) scaling limit, which we are calling the fixed irreducible bridge ensemble, for self-avoiding walks, and we conjecture a relationship between the fixed irreducible bridge ensemble and chordal SLE8/3 in the unit strip { z ∈ H : 0 < Im(z) < 1}.