Date & Time: Wednesday, August 17, 2016, 16:00-17:00.
Venue: Ramanujan Hall
Speaker: Parthanil Roy,
Indian Statistical Institute
Title: Extreme value theory for stable random fields indexed by finitely generated free groups
Abstract: In this work, we investigate the extremal behaviour of left-stationary \emph{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 (in the Cayley graph) of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying quasi-invariant 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 stable random fields generated by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a new class on point processes that we have termed as \emph{randomly thinned cluster Poisson processes}. This limit too is very different from that in the case of a lattice. This talk is based on a joint work with Sourav Sarkar, who carried out a significant portion of the work in his master's dissertation at Indian Statistical Institute.