Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties, such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that nontrivially satisfy the conditions for having this equivalence and a nonempty range of inverse temperatures where uniqueness holds.

Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu $-stationary probability measures on $G/\Lambda $ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on $G/\Lambda $ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda $ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.

Let $S=\{p_1, \ldots , p_r,\infty \}$ for prime integers $p_1, \ldots , p_r.$ Let X be an S-adic compact nilmanifold, equipped with the unique translation-invariant probability measure $\mu .$ We characterize the countable groups $\Gamma $ of automorphisms of X for which the Koopman representation $\kappa $ on $L^2(X,\mu )$ has a spectral gap. More specifically, let Y be the maximal quotient solenoid of X (thus, Y is a finite-dimensional, connected, compact abelian group). We show that $\kappa $ does not have a spectral gap if and only if there exists a $\Gamma $-invariant proper subsolenoid of Y on which $\Gamma $ acts as a virtually abelian group,

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g.

Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a q th power in $\Gamma _{g}$ , then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$

as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

We prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we show that the almost sure and $L^1$ convergences of the Shannon–McMillan–Breiman theorem hold for compactly supported random walks on compactly generated groups with subexponential growth.

Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$, and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $\mu $. Under the assumptions that $\mu $ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry–Esseen bound with the optimal rate $O(1/\sqrt n)$ for the coefficients of $S_n$, settling a long-standing question considered since the fundamental work of Guivarc’h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.

We study an example of a hit-and-run random walk on the symmetric group $\mathbf S_n$ . Our starting point is the well-understood top-to-random shuffle. In the hit-and-run version, at each single step, after picking the point of insertion j uniformly at random in $\{1,\ldots,n\}$ , the top card is inserted in the jth position k times in a row, where k is uniform in $\{0,1,\ldots,j-1\}$ . The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of n). Analyzing this problem in total variation is an interesting open question. We show that, in general, hit-and-run random walks on finite groups have non-negative spectrum.

A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec $ on the irreducible representations of ${{S}_{n}}$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the “Aldous order” completely is a generalized question. We show a few additional entries for this order.

We consider the stochastic difference equation ${{\eta }_{k}}\,=\,{{\xi }_{k}}\phi \left( {{\eta }_{k-1}} \right),\,\,k\,\in \,\mathbb{Z}$ on a locally compact group $G$, where $\phi $ is an automorphism of $G$, ${{\xi }_{K}}$ are given $G$-valued random variables, and ${{\eta }_{k}}$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ${{\xi }_{K}}$ have a common law $\mu $ and prove that if $G$ is a distal group and $\phi $ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu $ is supported on $Kz\,=\,z\phi \left( K \right)$ for any $z$ in the support of $\mu $. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

Let $G$ be a locally compact group and $\pi $ a representation of $G$ by weakly* continuous isometries acting in a dual Banach space $E$. Given a probability measure $\mu $ on $G$, we study the Choquet–Deny equation $\pi (\mu )x\,=\,x,\,x\,\in \,E$. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu $. The relation between the space of solutions of the Choquet–Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the ${{W}^{*}}$-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space $E\,=\,B({{L}^{2}}(G))$ of bounded linear operators on ${{L}^{2}}(G)$. Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

Suppose we are given the free product V of a finite family of finite or countable sets (Vi)i∈∮ and probability measures on each Vi, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vi. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup ${{\left( {{\mu }_{t}} \right)}_{t>0}}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if ${{\left( {{\mu }_{t}} \right)}_{t>0}}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto {{\mu }_{t}}\left( x \right),t>0$, such that ${{\lim }_{t\to 0}}t\log {{\mu }_{t}}\left( e \right)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega $ can be represented by a continuous function in $\Omega $. Examples are discussed.

We study the analysis of a probability density $K$ on a Lie group $G$, where $G$ is a semidirect product of a compact group $M$ with a nilpotent group $N$. To approximate analysis on $G$ with analysis on $N$, it is natural to consider certain maps (“realizations”) of $G$ onto $N$. In this paper, we prove the existence of a realization of $G$ in $N$ which is $K$-harmonic (modulo the commutator subgroup of $N$). By utilizing this result and extending some ideas of Alexopoulos, we can prove the boundedness in $L^p$ spaces of some new Riesz transforms associated with $K$, and obtain new regularity estimates for the convolution powers of $K$.

For a finite group $H$, let $Irr(H)$ denote the set of irreducible characters of $H$, and define the ‘zeta function’ $\zeta^H(t) = \sum_{\chi \in Irr(H)} \chi(1)^{-t}$ for real $t > 0$. We study the asymptotic behaviour of $\zeta^H(t)$ for finite simple groups $H$ of Lie type, and also of a corresponding zeta function defined in terms of conjugacy classes. Applications are given to the study of random walks on simple groups, and on base sizes of primitive permutation groups.