STABLE RANDOM FIELDS INDEXED BY FINITELY GENERATED FREE GROUPS

• 成果类型: Article
• 署名作者: Sarkar, Sourav; Roy, Parthanil
• 署名单位: University of California System; University of California Berkeley; Indian Statistical Institute; Indian Statistical Institute Bangalore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1236
• 发表日期: 2018
• 页码: 2680-2714
• 关键词: infinitely divisible processes; branching random-walks; self-similar processes; central-limit-theorem; ergodic properties; point-processes; conservative flows; long memory; REPRESENTATIONS; BOUNDARY

摘要

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 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 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.