52 research outputs found
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
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
Thompson's group is not Liouville
We prove that random walks on Thompson's group driven by strictly
non-degenerate finitely supported probability measures have a non-trivial
Poisson boundary. The proof consists in an explicit construction of two
different non-trivial -boundaries. Both of them are defined in terms of
the Schreier graph on the dyadic-rational orbit of the canonical
action of on the unit interval (actually, we consider a natural embedding
of into the group of piecewise linear homeomorphisms of
the real line, and realize on the dyadic-rational orbit in ). However, the behaviours at infinity described by these -boundaries
are quite different (in perfect keeping with the ambivalence concerning
amenability of the group ). The first -boundary is similar to the
boundaries of the lamplighter groups: it consists of -valued
configurations on arising from the stabilization of the logarithmic
increments of slopes along the sample paths of the random walk. The second
-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
: these are the collections of the limit ends of the induced random
walk on parameterized by all possible starting points
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
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
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
Circular slider graphs: de Bruijn, Kautz, Rauzy, lamplighters and spiders
We suggest a new point of view on de Bruijn graphs and their subgraphs based
on using circular words rather than linear ones
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 of a free group on the boundary
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 associated with a geodesic spanning tree in the Schreier
graph . 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 and of the subgroup ( cogrowth
of ), respectively. We also construct numerous examples illustrating
connections between various relevant notions.Comment: minor editorial changes, added references; final version to appear in
Advances in Mathematic
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
Matrix random products with singular harmonic measure
Any Zariski dense countable subgroup of 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 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
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