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September 2018 Stable random fields indexed by finitely generated free groups
Sourav Sarkar, Parthanil Roy
Ann. Probab. 46(5): 2680-2714 (September 2018). DOI: 10.1214/17-AOP1236


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.


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Sourav Sarkar. Parthanil Roy. "Stable random fields indexed by finitely generated free groups." Ann. Probab. 46 (5) 2680 - 2714, September 2018.


Received: 1 August 2016; Revised: 1 August 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964346
MathSciNet: MR3846836
Digital Object Identifier: 10.1214/17-AOP1236

Primary: 60G52 , 60G60
Secondary: 20E05 , 37A40 , 60G55

Keywords: boundary action , Extreme value theory , free group , nonsingular group action , point process , Random field , Stable

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 5 • September 2018
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