Related papers: Equidistribution of affine random walks on some ni…

We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk under the assumption that the Zariski closure of the group generated…

We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

Let $\mu$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, \[ \left\{\begin{array}{rcl} X_0…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense…

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/\Gamma$ is a nilmanifold, $a_1,\ldots,a_k\in G$ are commuting nilrotations, and…

The main goal of this paper is to obtain optimal estimates on the speed of equidistribution of nilflows on higher step nilmanifolds. Under a Diophantine condition on the frequencies of the toral projection of the flow, we prove that for…

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the $n$-step distributions from almost every starting point…

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for…

We prove quantitative equidistribution results for actions of Abelian subgroups of the $2g+1$ dimensional Heisenberg group acting on compact $2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study of the…

Let $\rho$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and consider the random walk defined by $\rho$ on the torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved that under an…

We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with H\"older continuous test functions $f$, including the…

In this paper, we rigorously establish the Gumbel-distributed fluctuations of the cover time, normalized by the mean first passage time, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. This has…

Generalizing classic results for a family of measures in the torus, for a family $(\mu_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study conditions under which the family equidistributes, meaning conditions under which the…

The main results of this paper are to prove bounds for ergodic averages for nilflows on general higher step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become…

Let $X_1, X_2, \dots$ be independent, identically distributed random variables taking values from a compact metrizable group $G$. We prove that the random walk $S_k=X_1 X_2 \cdots X_k$, $k=1,2,\dots$ equidistributes in any given Borel…

The unstable foliation, that locally is given by changing horizontal components of period coordinates, plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup.…

Let $G$ be a real Lie group, $\Lambda\subseteq G$ a lattice, and $X=G/\Lambda$. We fix a probability measure $\mu$ on $G$ and consider the left random walk induced on $X$. It is assumed that $\mu$ is aperiodic, has a finite first moment,…

We formulate and establish the central limit theorem for products of i.i.d. random variables on arbitrary simply connected nilpotent Lie groups, allowing a possible bias. Two new phenomena arise in the presence of a bias: (a) the walk…