Stable random fields, Patterson-Sullivan measures and extremal cocycle growth

• 成果类型: Article
• 署名作者: Athreya, Jayadev S.; Mj, Mahan; Roy, Parthanil
• 署名单位: University of Washington; University of Washington Seattle; Tata Institute of Fundamental Research (TIFR); Indian Statistical Institute; Indian Statistical Institute Bangalore
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01134-z
• 发表日期: 2022
• 页码: 681-714
• 关键词: quasi-conformal measures; cannon-thurston maps; ergodic-theory; equidistribution; transformations; BOUNDARY; sets

摘要

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, and (3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric a-stable (0 < alpha < 2) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.