Large deviations of convex hulls of planar random walks and Brownian motions

TL;DR

This paper establishes large deviations principles for the perimeter and area of convex hulls of planar random walks and Brownian motions, providing explicit rate functions and identifying optimal trajectory shapes for various distributions.

Contribution

It derives explicit large deviations rate functions for convex hull perimeter and area, including for Gaussian, rotationally invariant, and Levy process increments, with detailed shape characterizations.

Findings

Rate functions for perimeter and area are explicitly derived.

Optimal trajectories align into line segments or arcs depending on the distribution.

Results extend to Brownian motions and Levy processes with finite Laplace transforms.

Abstract

We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general. Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise. The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Levy processes with finite Laplace transform.