24 research outputs found
A Spectral Strong Approximation Theorem for Measure Preserving Actions
Let be a finitely generated group acting by probability measure
preserving maps on the standard Borel space . We show that if
is a subgroup with relative spectral radius greater than the
global spectral radius of the action, then acts with finitely many ergodic
components and spectral gap on . This answers a question of Shalom who
proved this for normal subgroups.Comment: 17 page
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Arbeitsgemeinschaft: Limits of Structures
The goal of the Arbeitsgemeinschaft is to review current progress in the study of very large structures. The main emphasis is on the analytic approach that considers large structures as approximations of infinite analytic objects. This approach enables one to study graphs, hypergraphs, permutations, subsets of groups and many other fundamental structures
Factor-of-iid balanced orientation of non-amenable graphs
We show that if a non-amenable, quasi-transitive, unimodular graph has
all degrees even then it has a factor-of-iid balanced orientation, meaning each
vertex has equal in- and outdegree. This result involves extending earlier
spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of
quasi-transitive, unimodular graphs. As a consequence, we also obtain that when
is regular (of either odd or even degree) and bipartite, it has a
factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov
beyond transitive graphs.Comment: 24 pages, 1 figure. This is one of two papers that are replacing the
shorter arXiv submission arXiv:2101.12577v1 Factor of iid Schreier decoration
of transitive graph
Factors of IID on Trees
Classical ergodic theory for integer-group actions uses entropy as a complete
invariant for isomorphism of IID (independent, identically distributed)
processes (a.k.a. product measures). This theory holds for amenable groups as
well. Despite recent spectacular progress of Bowen, the situation for
non-amenable groups, including free groups, is still largely mysterious. We
present some illustrative results and open questions on free groups, which are
particularly interesting in combinatorics, statistical physics, and
probability. Our results include bounds on minimum and maximum bisection for
random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur
Correlation bound for distant parts of factor of IID processes
We study factor of i.i.d. processes on the -regular tree for .
We show that if such a process is restricted to two distant connected subgraphs
of the tree, then the two parts are basically uncorrelated. More precisely, any
functions of the two parts have correlation at most ,
where denotes the distance of the subgraphs. This result can be considered
as a quantitative version of the fact that factor of i.i.d. processes have
trivial 1-ended tails.Comment: 18 pages, 5 figure
A spectral strong approximation theorem for measure-preserving actions
Let be a finitely generated group acting by probability measure-preserving maps on the standard Borel space. We show that if is a subgroup with relative spectral radius greater than the global spectral radius of the action, then acts with finitely many ergodic components and spectral gap on. This answers a question of Shalom who proved this for normal subgroups. © Cambridge University Press, 2018
Correlation bounds for distant parts of factor of IID processes
We study factor of i.i.d. processes on the d-regular tree for d ≥ 3. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most , where k denotes the distance between the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails