Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

By S. Gouëzel. 2012

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Consider a countable group Γ (with identity denoted by e), together with a probability measure μ whose support generates Γ as a semigroup (we say that μ is admissible). Multiplying random elements of Γ distributed independently according to μ, one obtains a random walk on Γ. The local limit problem consists in determining good asymptotics for the transition probabilities pn(x, y) of this random walk. Let us assume for simplicity that μ is finitely supported. For Γ = Z, simple Fourier computations show that pn(e, e) ∼ Cn−d/2 if the walk is centered, and pn(e, e) ∼ CR−nn−d/2 for some R > 1 if the walk is not centered. Similar asymptotics hold in nilpotent groups by the deep results of Varopoulos and Alexopoulos [Ale02]. When the group is not amenable, pn(e, e) decays exponentially fast. The situation is well understood for semisimple Lie groups and absolutely continuous measures since the work of Bougerol [Bou81]: the probability to return to a fixed neighborhood of the identity behaves like CR−nn−a for some R > 1 (depending on the measure one considers) and some a > 1 only depending on the geometry of the group. In the simplest case of rank one groups, a = 3/2. It is reasonable to conjecture that similar asymptotics (with the same a) hold for random walks on cocompact lattices of such semisimple Lie groups, but the proofs of Bougerol (based on representation theory) do not adapt well, and this question is essentially open. A notable exception is the case of free groups: in this situation, the generating function of the transition probabilities (also called the Green function) Gr(x, y) = ∑ rpn(x, y) is an algebraic function of r. A careful study of its first singularity then yields the asymptotics of pn(x, y). For free groups, this is due to Lalley [Lal93], and the asymptotics is of the form pn(x, y) ∼ C(x, y)R−nn−3/2, in accordance with the results of Bougerol in rank one Lie groups. Most free products can also be treated similarly; see [Woe00, Chapter III] and references therein. Recently, together with Lalley, we were able to treat in [GL13] some nonamenable groups where the Green function is not expected to be algebraic. We proved that, for a cocompact lattice of PSL(2,R), and for a finitely supported symmetric measure μ, the above asymptotics pn(x, y) ∼ C(x, y)R−nn−3/2 still holds. Henceforth, we will refer to this situation as the PSL(2,R)-case. The overall strategy can in fact be

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

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