106 research outputs found
Perfect Matchings as IID Factors on Non-Amenable Groups
We prove that in every bipartite Cayley graph of every non-amenable group,
there is a perfect matching that is obtained as a factor of independent uniform
random variables. We also discuss expansion properties of factors and improve
the Hoffman spectral bound on independence number of finite graphs.Comment: 16 pages; corrected missing reference in v
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
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
Baire Measurable Matchings in Non-Amenable Graphs
We prove that every Schreier graph of a free Borel action of a finitely
generated non-amenable group has a Baire measurable perfect matching. This
result was previously only known in the bipartite setting. We also prove that
every Borel non-amenable bounded degree graph with only even degrees has a
Baire measurable balanced orientation
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
Baire measurable paradoxical decompositions via matchings
We show that every locally finite bipartite Borel graph satisfying a
strengthening of Hall's condition has a Borel perfect matching on some comeager
invariant Borel set. We apply this to show that if a group acting by Borel
automorphisms on a Polish space has a paradoxical decomposition, then it admits
a paradoxical decomposition using pieces having the Baire property. This
strengthens a theorem of Dougherty and Foreman who showed that there is a
paradoxical decomposition of the unit ball in using Baire
measurable pieces. We also obtain a Baire category solution to the dynamical
von Neumann-Day problem: if is a nonamenable action of a group on a Polish
space by Borel automorphisms, then there is a free Baire measurable action
of on which is Lipschitz with respect to .Comment: Minor revision
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