258 research outputs found
A Structure Theorem for Small Sumsets in Nonabelian Groups
Let G be an arbitrary finite group and let S and T be two subsets such that
|S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1
then either S is a geometric progression or there exists a non-trivial subgroup
H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the
nonabelian case classical results for Abelian groups. When we remove the
hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the
above characterization whose structure is described precisely.Comment: 23 page
Multi-way expanders and imprimitive group actions on graphs
For n at least 2, the concept of n-way expanders was defined by various
researchers. Bigger n gives a weaker notion in general, and 2-way expanders
coincide with expanders in usual sense. Koji Fujiwara asked whether these
concepts are equivalent to that of ordinary expanders for all n for a sequence
of Cayley graphs. In this paper, we answer his question in the affirmative.
Furthermore, we obtain universal inequalities on multi-way isoperimetric
constants on any finite connected vertex-transitive graph, and show that gaps
between these constants imply the imprimitivity of the group action on the
graph.Comment: Accepted in Int. Math. Res. Notices. 18 pages, rearrange all of the
arguments in the proof of Main Theorem (Theorem A) in a much accessible way
(v4); 14 pages, appendix splitted into a forthcoming preprint (v3); 17 pages,
appendix on noncommutative L_p spaces added (v2); 12 pages, no figure
Convergent expansions for Random Cluster Model with q>0 on infinite graphs
In this paper we extend our previous results on the connectivity functions
and pressure of the Random Cluster Model in the highly subcritical phase and in
the highly supercritical phase, originally proved only on the cubic lattice
, to a much wider class of infinite graphs. In particular, concerning the
subcritical regime, we show that the connectivity functions are analytic and
decay exponentially in any bounded degree graph. In the supercritical phase, we
are able to prove the analyticity of finite connectivity functions in a smaller
class of graphs, namely, bounded degree graphs with the so called minimal
cut-set property and satisfying a (very mild) isoperimetric inequality. On the
other hand we show that the large distances decay of finite connectivity in the
supercritical regime can be polynomially slow depending on the topological
structure of the graph. Analogous analyticity results are obtained for the
pressure of the Random Cluster Model on an infinite graph, but with the further
assumptions of amenability and quasi-transitivity of the graph.Comment: In this new version the introduction has been revised, some
references have been added, and many typos have been corrected. 37 pages, to
appear in Communications on Pure and Applied Analysi
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
Explicit isoperimetric constants and phase transitions in the random-cluster model
The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model on nonamenable graphs with
cluster parameter . Among these, the main ones are the absence of
percolation for the free random-cluster measure at the critical value, and
examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w
(q) for large enough. The latter follows from considerations of
isoperimetric constants, and we give the first nontrivial explicit calculations
of such constants. Such considerations are also used to prove non-robust phase
transition for the Potts model on nonamenable regular graphs
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