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Vadim A. Kaimanovich

Publications and source records attributed to Vadim A. Kaimanovich.

16 recordsLinked to original sources
arXiv2022

Limit distributions of branching Markov chains

We study branching Markov chains on a countable state space (space of types) $\mathscr{X}$, with the focus on the qualitative aspects of the limit behaviour of the evolving empirical population distributions. No conditions are imposed on the multitype offspring distributions at the points of $\mathscr{X}$ other than to have the same average and to satisfy a uniform $L \log L$ moment condition. We show that the arising population martingale is uniformly integrable. Convergence of population averages of the branching chain is then put in connection with stationary spaces of the associated ordinary Markov chain on $\mathscr{X}$ (assumed to be irreducible and transient). This is applied, in particular, to the boundaries of appropriate compactifications of $\mathscr{X}$. Final considerations consider the general interplay between the measure theoretic boundaries of the branching chain and the associated ordinary chain.

math.PR↗
arXiv2020

Amenability of groupoids and asymptotic invariance of convolution powers

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich-Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

math.FA↗
arXiv2016

Thompson's group $F$ is not Liouville

We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). However, the behaviours at infinity described by these $μ$-boundaries are quite different (in perfect keeping with the ambivalence concerning amenability of the group $F$). The first $μ$-boundary is similar to the boundaries of the lamplighter groups: it consists of ${\mathbb Z}$-valued configurations on $Γ$ arising from the stabilization of the logarithmic increments of slopes along the sample paths of the random walk. The second $μ$-boundary is more similar to the boundaries of groups with hyperbolic properties as it consists of the sections of the end bundle of the graph $Γ$: these are the collections of the limit ends of the induced random walk on $Γ$ parameterized by all possible starting points.

math.GR↗
arXiv2015

Invariance, quasi-invariance and unimodularity for random graphs

We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in considering quasi-invariant (rather than just invariant) measures with respect to the root moving equivalence relation. We define a natural modular cocycle of this equivalence relation, and show that unimodular measures are precisely those quasi-invariant measures whose Radon--Nikodym cocycle coincides with the modular cocycle. This embeds the notion of unimodularity into the very general dynamical scheme of constructing and studying measures with a prescribed Radon--Nikodym cocycle.

math.PR↗
arXiv2012

Random walks on random horospheric products

By developing the entropy theory of random walks on equivalence relations and analyzing the asymptotic geometry of horospheric products we describe the Poisson boundary for random walks on random horospheric products of trees.

math.PR↗
arXiv2011

Ergodic properties of boundary actions and Nielsen--Schreier theory

We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup $H$ of a free group $F$ on the boundary $\partial F$ with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen--Schreier generators in $H$ associated with a geodesic spanning tree in the Schreier graph $X=H\backslash F$. We give several (mod 0) equivalent descriptions of the Hopf decomposition of the boundary into the conservative and the dissipative parts. Further we relate conservativity and dissipativity of the action with the growth of the Schreier graph $X$ and of the subgroup $H$ ($\equiv$ cogrowth of $X$), respectively. We also construct numerous examples illustrating connections between various relevant notions.

math.GR↗
arXiv2009

Stochastic homogenization of horospheric tree products

We construct measures invariant with respect to equivalence relations which are graphed by horospheric products of trees. The construction is based on using conformal systems of boundary measures on treed equivalence relations. The existence of such an invariant measure allows us to establish amenability of horospheric products of random trees.

math.PR↗
arXiv2009

Self-similarity and random walks

This is an introductory level survey of some topics from a new branch of fractal analysis -- the theory of self-similar groups. We discuss recent works on random walks on self-similar groups and their applications to the problem of amenability for these groups.

math.GR↗
arXiv2008

Hopf decomposition and horospheric limit sets

By looking at the relationship between the recurrence properties of a countable group action with a quasi-invariant measure and the structure of its ergodic components we establish a simple general description of the Hopf decomposition of the action into the conservative and the dissipative parts in terms of the Radon--Nikodym derivatives of the action. As an application we prove that the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to any invariant quasi-conformal stream coincides (mod 0) with the big horospheric limit set of the group.

math.DS↗
arXiv2008

Matrix random products with singular harmonic measure

Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.

math.PR↗
arXiv2008

On amenability of automata groups

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups ("Mother groups") which is done by analyzing the asymptotic properties of random walks on these groups.

math.GR↗
arXiv2004

Boundaries and harmonic functions for random walks with random transition probabilities

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.

math.PR↗
arXiv2004

Amenability and the Liouville property

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid action on the Poisson boundary of any invariant operator is always amenable. This approach subsumes as particular cases numerous earlier results on amenability for groups, actions, equivalence relations and foliations. For instance, we establish in a unified way topological amenability of the boundary action for isometry groups of Gromov hyperbolic spaces, Riemannian symmetric spaces and affine buildings.

math.FA↗
arXiv2004

Boundary amenability of hyperbolic spaces

It is well-known that a Kleinian group is amenable if and only if it is elementary. We establish an analogous property for equivalence relations and foliations with Gromov hyperbolic leaves: they are amenable if and only if they are elementary in the sense that one can assign (in a measurable way) to any leaf a finite subset of its hyperbolic boundary (as in the group case, such subsets cannot actually contain more than 2 points). The analogous result for actions of word hyperbolic groups with a quasi-invariant measure is that such an action is amenable if and only if it factorizes through the hyperbolic boundary or its symmetric square. A byproduct of our approach is a proof of boundary amenability for isometry groups of Gromov hyperbolic spaces under the assumption that either the space is exponentially bounded or it is CAT(-1) and the group has a finite critical exponent. We also give examples showing that without these assumptions the boundary amenability may fail.

math.FA↗
arXiv2000

The Poisson formula for groups with hyperbolic properties

The Poisson boundary of a group G with a probability measure μis the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded μ-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semi-simple Lie groups.

math.DS↗