Large Deviations of Convex Hulls of the "True" Self-Avoiding Random Walk

TL;DR

This paper investigates the distribution of the area and perimeter of convex hulls of the true self-avoiding random walk, using advanced sampling to explore rare events and test theoretical predictions about their scaling and large deviations.

Contribution

It provides the first detailed analysis of large deviations for convex hull properties of the true self-avoiding walk, confirming some scaling conjectures and revealing deviations in the rate function behavior.

Findings

Distribution governed by Flory exponent with logarithmic corrections

Confirmed scaling conjectures for the distribution

Observed deviations in the large-deviation rate function

Abstract

We study the distribution of the area and perimeter of the convex hull of the "true" self-avoiding random walk in a plane. Using a Markov chain Monte Carlo sampling method, we obtain the distributions also in their far tails, down to probabilities like $10^{-800}$. This enables us to test previous conjectures regarding the scaling of the distribution and the large-deviation rate function $\Phi$. In previous studies, e.g., for standard random walks, the whole distribution was governed by the Flory exponent $\nu$. We confirm this in the present study by considering expected logarithmic corrections. On the other hand, the behavior of the rate function deviates from the expected form. For this exception we give a qualitative reasoning.