Ergodic theory of affine isometric actions on Hilbert spaces

TL;DR

This paper extends Gaussian actions to affine isometric actions on Hilbert spaces, revealing a phase transition phenomenon and connecting it to geometric invariants, with applications to groups acting on trees.

Contribution

It introduces a new construction linking affine isometric actions to Gaussian actions and analyzes their ergodic phase transitions using geometric and probabilistic tools.

Findings

Identification of a phase transition in nonsingular Gaussian actions

Connection between ergodic properties and geometry of affine actions

Existence of free, weakly mixing Gaussian actions of stable type III₁ for groups without property (T)

Abstract

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type $\mathrm{III}_1$.