Stable random fields indexed by finitely generated free groups

TL;DR

This paper studies the extremal behavior of symmetric alpha-stable random fields indexed by free groups, revealing a phase transition influenced by ergodic properties and introducing new limit processes for maxima.

Contribution

It introduces a phase transition in extremal growth depending on ergodic properties of free group actions and characterizes new limit processes for maxima in this setting.

Findings

Discovered a phase transition in maxima growth related to ergodic properties.

Identified a conservative action case with maxima growing as in i.i.d. sequences.

Established convergence of extremal point processes to novel thinned cluster Poisson processes.

Abstract

In this work, we investigate the extremal behaviour of left-stationary symmetric $\alpha$-stable (S$\alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of S$\alpha$S random fields indexed by $\mathbb{Z}^d$. The presence of this new dichotomy is confirmed by the study of a stable random field induced by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. This field is generated by a conservative action but its maxima sequence grows as fast as the i.i.d. case contrary to what happens in the case of $\mathbb{Z}^d$. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a novel class of point processes that we have termed as randomly thinned cluster Poisson processes. This limit too is very different from that in the case of a lattice.