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

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.