Random walks on hyperbolic spaces: Concentration inequalities and probabilistic Tits alternative

Abstract

The goal of this article is two-fold: in a first part, we prove Azuma–Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space M, we obtain explicit bounds that depend only on M, the size of support of the measure as in the classical case of sums of independent random variables, and on the norm of the driving probability measure in the left regular representation of the group of isometries. We obtain uniform bounds in the case of hyperbolic groups and effective bounds for simple linear groups of rank-one. In a second part, using our concentration inequalities, we give quantitative finite-time estimates on the probability that two independent random walks on the isometry group of a hyperbolic space generate a free non-abelian subgroup. Our concentration results follow from a more general, but less explicit statement that we prove for cocycles which satisfy a certain cohomological equation. For example, this also allows us to obtain subgaussian concentration bounds around the top Lyapunov exponent of random matrix products in arbitrary dimension.

Random walks on hyperbolic spaces: second order expansion of the rate function at the drift

“…4.3.1] for every λ < α, we have I * (λ) = Λ(λ), where I * is the Fenchel-Legendre transform of I. But since the second-order term σ 2 µ /2 in the second-order expansion of Λ at 0 (given by Theorem 1.1) is positive, it follows that the Fenchel-Legendre transforms of I * and Λ coincides in a neighborhood of ℓ µ , i.e., 1) . This is in line with the recent work [1] where, under additional assumptions, the appearing constants are made explicit (e.g.…”

“…It was then verified in [22] that this solution can be extended to the case when X is non-proper and also in [1] that ψ could be defined on the whole compactification X h while preserving the boundedness of ψ. This is equivalent to finding a cocycle σ 0 : G × X h → R with constant drift equal to ℓ µ , i.e., σ 0 (g, x)dµ(g) = ℓ µ for every x ∈ X h , such that the following identity holds for every…”

“…The first one (discussed in Section 3.2.1) is to recall a martingale decomposition (Lemma 3.1) of the Busemann cocycle along non-elementary random walks on Gromov-hyperbolic spaces which is due to Benoist-Quint [2,3] (see also an extension in [22]). We will use a slightly more general version of this worked out in [1]. The second goal (discussed in Section 3.2.2) is to prove Proposition 3.3 about large deviations of predictable quadratic variation and its consequence expressed in Corollary 3.9.…”

“…where int(R) denotes the interior and R the closure of R. Furthermore, concentration inequalities reminiscent of Hoeffding inequalities were recently shown by Aoun-Sert [1] and a local limit theorem for random walks on Gromov-hyperbolic groups was proved by Gouëzel [18]. However, establishing these results analogous to the classical setting of sums of i.i.d.…”

“…We will extensively use the martingale approach -developed in this setting by Benoist-Quint [2,3] and adapted to greater generality by Horbez [22] and Aoun-Sert [1] -to tackle the analytic problem of giving a second-order expansion of the limit Laplace transform of the sequence d(z n , o) (or by convex duality, of its large deviation rate function in (1.3)) and relating it to the variance in the central limit theorem (1.2). Similar results are known to hold in settings where spectral methods are available.…”