[Paper Review] Large deviations of convex hulls of planar random walks

Arseniy Akopyan, Vladislav Vysotsky|arXiv (Cornell University)|Jun 23, 2016

Point processes and geometric inequalities25 references4 citations

TL;DR

This paper establishes large deviations principles for the perimeter and area of the convex hull of planar random walks with finite Laplace transforms on increments. It derives explicit rate functions for perimeter and area, showing that for rotationally symmetric or Gaussian increments, optimal trajectories align into line segments (perimeter) or half-circles (area), while for Gaussian walks with drift, optimal shapes are elliptic or parabolic arcs depending on the covariance structure.

ABSTRACT

We prove large deviations principles 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, hence giving 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 of the convex hull are attained by the trajectories that asymptotically align into line segments but in general, line segments may not be optimal. 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 found 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.

Motivation & Objective

• To establish large deviations principles for the perimeter and area of the convex hull of planar random walks.
• To characterize the rate functions of these functionals under general conditions on the increment distribution.
• To identify the optimal trajectory shapes that realize large deviations for perimeter and area.
• To determine explicit expressions for the rate function in special cases, including Gaussian and rotationally invariant increments.
• To analyze the dependence of optimal shapes on the mean and covariance of the increments.

Proposed method

• The analysis uses large deviations theory applied to the convex hull of planar random walks with increments having finite Laplace transforms.
• The rate function for the perimeter is bounded above and below in terms of the increment distribution's rate function, with equality achieved for rotationally symmetric and Gaussian increments.
• For area, explicit rate functions are derived using symmetry and moment-generating function techniques, particularly for rotationally invariant and Gaussian walks with drift.
• Optimal trajectories are identified via variational principles, showing that large deviations are realized by paths asymptotically aligning into line segments, half-circles, or elliptic/parabolic arcs.
• The method distinguishes between zero-mean and non-zero-mean increments, with different optimal shapes emerging in each case.
• The derivation leverages properties of convex hulls and the geometry of random walk paths under extreme deviations.

Experimental results

Research questions

• RQ1 What is the rate function for the perimeter of the convex hull of a planar random walk with finite Laplace transform on increments?
• RQ2 How do the optimal trajectories that realize large deviations of the perimeter depend on the distribution of increments?
• RQ3 What is the rate function for the area of the convex hull in the case of rotationally invariant increments?
• RQ4 How do the optimal shapes for large deviations of the area change when the increments have non-zero mean and non-degenerate covariance?
• RQ5 Under what conditions do line segments, half-circles, or elliptic/parabolic arcs become the optimal limiting shapes for convex hull functionals?

Key findings

• For random walks with rotationally symmetric or Gaussian increments, the rate function for the perimeter is explicitly determined by matching upper and lower bounds derived from the increment distribution's rate function.
• The optimal trajectories for large deviations of the perimeter are asymptotically aligned into line segments, though such shapes are not always optimal in general.
• For zero-mean, rotationally invariant increments, the rate function of the area is explicitly computed, and optimal trajectories align into half-circles.
• For Gaussian walks with drift and non-degenerate covariance, the optimal shape for large deviations of the area is an elliptic arc.
• When the covariance matrix of the increments is degenerate, the optimal shape for area deviations becomes a parabolic arc.
• The results establish a complete characterization of large deviations for both perimeter and area, with shape optimality depending on the mean and second-order structure of the increments.