Loop-Erased Random Walk as a Spin System Observable
- PDF / 400,625 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 48 Downloads / 133 Views
Loop-Erased Random Walk as a Spin System Observable Tyler Helmuth1
· Assaf Shapira2
Received: 2 April 2020 / Accepted: 17 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk. Keywords Loop-erased random walk · Spin system · Heaps of pieces · Random walk representation
1 Introduction Loop-erased random walk is, informally speaking, the probability measure on self-avoiding walks that results from removing the loops from simple random walk in chronological order. We give a precise description in Sect. 2.2 below. Loop-erased random walk is a fundamental probabilistic object with connections to spanning trees and the uniform spanning forest [20,26], amongst other topics. In two dimensions, loop-erased random walk has SLE2 as a scaling limit [15], while in four and higher dimensions it scales to Brownian motion [14]. It is possible to prove the scaling limit of loop-erased random walk exists in three dimensions [12], but many open questions remain, see [3,16] and references therein. The preceding results have been used to determine the fractal (Hausdorff) dimension dimH (Kd ) of the scaling limit Kd of d-dimensional loop-erased random walk when d = 3:
Communicated by Yvan Velenik.
B
Tyler Helmuth [email protected] Assaf Shapira [email protected]
1
Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK
2
Università Roma Tre, Roma, Italy
123
T. Helmuth, A. Shapira
dimH (K2 ) = 45 and dimH (Kd ) = 2 for d ≥ 4. Shiraishi [22] has given a characterization of dimH (K3 ), but the numerical value is not known rigorously. These results are all based on probabilistic tools. Loop-erased random walk is also of interest within theoretical physics. An interesting recent development has been the use of non-rigorous field theory techniques for the determination of the Hausdorff dimension dimH (Kd ) [8,25]. While there is a long history of the interplay between field theories and random walks [6,7,23], geometric properties of looperased random walk are less obviously connected to a field theory due to ‘erasure’ in its definition. In this note we describe two rigorous spin system representations of loop-erased random walk. Both representations translate the problem of the determination of the Hausdorff dimension for loop-erased random walk in
Data Loading...
