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 SLE₈/₃. 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 SLE₈/₃. In [DGKLP2011], 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 ℙ(α), 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 [AC2010]. 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 [LSW2002] 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}.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/306346 |
| Date | January 2013 |
| Creators | Gilbert, Michael James |
| Contributors | Kennedy, Tom, Kennedy, Tom, Pickrell, Douglas, Wehr, Janek, Sethuraman, Sunder |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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