A series test of the scaling limit of self-avoiding walks

TL;DR

This paper tests the conjecture that the scaling limit of self-avoiding walks in rectangles conforms to SLE with a specific parameter, by calculating hitting probabilities and estimating the value of .

Contribution

It provides the first precise numerical estimates of  for self-avoiding walks in rectangles, supporting the SLE conjecture and analyzing boundary hitting distributions.

Findings

Estimated   2.66664  0.00007 for aspect ratio 2

Estimated   2.66675  0.00015 for aspect ratio 10

Numerical evidence supports the boundary hitting distribution conjecture

Abstract

It is widely believed that the scaling limit of self-avoiding walks (SAWs) at the critical temperature is (i) conformally invariant, and (ii) describable by Schramm-Loewner Evolution (SLE) with parameter $\kappa = 8/3.$ We consider SAWs in a rectangle, which originate at its centre and end when they reach the boundary. We assume that the scaling limit of SAWs is describable by ${\rm SLE}_\kappa,$ with the value of $\kappa$ to be determined. It has previously been shown by Guttmann and Kennedy \cite{GK13} that, in the scaling limit, the ratio of the probability that a SAW hits the side of the rectangle to the probability that it hits the end of the rectangle, depends on $\kappa.$ By considering rectangles of fixed aspect ratio 2, and also rectangles of aspect ratio 10, we calculate the probabilities exactly for larger and larger rectangles. By extrapolating this data to infinite rectangle size, we obtain the estimate $\kappa = 2.66664 \pm 0.00007$ for rectangles of aspect ratio 2 and $\kappa = 2.66675 \pm 0.00015$ for rectangles of aspect ratio 10. We also provide numerical evidence supporting the conjectured distribution of SAWs striking the boundary at various points in the case of rectangles with aspect ratio 2.