541 research outputs found
Uniqueness and non-uniqueness in percolation theory
This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on and,
more generally, on transitive graphs. For iid percolation on ,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A nonamenable "factor" of a Euclidean space
Answering a question of Benjamini, we present an isometry-invariant random
partition of the Euclidean space , , into infinite
connected indistinguishable pieces, such that the adjacency graph defined on
the pieces is the 3-regular infinite tree. Along the way, it is proved that any
finitely generated one-ended amenable Cayley graph can be represented in
as an isometry-invariant random partition of to
bounded polyhedra, and also as an isometry-invariant random partition of
to indistinguishable pieces. A new technique is developed to
prove indistinguishability for certain constructions, connecting this notion to
factor of iid's.Comment: 23 pages, 4 figure
The speed of biased random walk among random conductances
We consider biased random walk among iid, uniformly elliptic conductances on
, and investigate the monotonicity of the velocity as a function
of the bias. It is not hard to see that if the bias is large enough, the
velocity is increasing as a function of the bias. Our main result is that if
the disorder is small, i.e. all the conductances are close enough to each
other, the velocity is always strictly increasing as a function of the bias,
see Theorem 1. A crucial ingredient of the proof is a formula for the
derivative of the velocity, which can be written as a covariance, see Theorem
3: it follows along the lines of the proof of the Einstein relation in [GGN].
On the other hand, we give a counterexample showing that for iid, uniformly
elliptic conductances, the velocity is not always increasing as a function of
the bias. More precisely, if and if the conductances take the values
(with probability ) and (with probability ) and is close
enough to and small enough, the velocity is not increasing as a
function of the bias, see Theorem 2
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
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