Harmonic measures versus quasiconformal measures for hyperbolic groups
By S'ebastien Blachere, Peter Haissinsky, Pierre Mathieu. 2008
132 citations recorded by the source corpus.
We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2026db6ad58e0affcb28e7928e6787559ffdae18
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Asymptotic entropy and Green speed for random walks on countable groups (2006, unknown): “In a forthcoming paper [5], we use the Green metric to study fine geometric properties of the harmonic measure on boundaries of hyperbolic groups.”
- The Ergodic Theory of Hyperbolic Groups (2011, background): “ow that the Green metric and the word metric are quasi-isometric (one needs to be somewhat careful: the Green metric is not in general a geodesic metric). Theorem 4.4.2 (Blach`ere–Ha¨ıssinsky–Mathieu [2], Thm. 1.3). Let Gbe a nonelementary hyperbolic group, and for y∈ ∂∞G, let B(y,R) denote the ball”
- Random walks on co-compact fuchsian groups (2013, unknown): “The Green metric associated with F1 has been used by a number of authors, for instance in [7], [8] and [9].”
- GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “ solves this question for lattices in PSL(2;R). However, this strongly relies on the geometry of the Poincare disk, and does not extend to general groups. Moreover, it is shown in [35, 60] (see also [8]) that the resulting probability measure is singular with respect to the Lebesgue measure for non ”