Stable Random Fields, Patterson-Sullivan measures and Extremal Cocycle Growth

TL;DR

This paper investigates the extreme values of stable random fields indexed by groups acting on various geometric spaces, linking their behavior to the group's action via Patterson-Sullivan measures and introducing an invariant called extremal cocycle growth.

Contribution

It introduces extremal cocycle growth as a new invariant connecting geometric group actions with extreme value theory, and establishes a dichotomy in the growth of maxima for stable random fields.

Findings

Non-vanishing extremal cocycle growth relates to finite Bowen-Margulis measure.

Dichotomy in maxima growth rates for stable random fields based on group actions.

Results extend to normal subgroups of free groups.

Abstract

We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space $X$ and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.