The isoperimetric problem for convex hulls and the large deviations rate functionals of random walks

TL;DR

This paper investigates the asymptotic behavior of large deviations in the convex hull area of planar random walks, linking it to an anisotropic isoperimetric problem and explicitly solving for optimal trajectories.

Contribution

It establishes a connection between large deviations of convex hulls and anisotropic isoperimetric problems, providing explicit solutions for the optimal trajectories.

Findings

Optimal trajectories are convex and satisfy Euler-Lagrange equations.

Trajectories resemble Minkowski plane isoperimetric optimizers.

Results depend on the rate function of the increments.

Abstract

We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite on $R^2$, such a scaled limit trajectory $h$ solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of $h$ is replaced by the large deviations rate functional $\int_0^1 I(h'(t)) dt$ and $I$ is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler-Lagrange equation, which we solve explicitly for every $I$. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).