16,236 research outputs found
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot have
infinitely many infinite clusters at critical Bernoulli percolation. In the
case of heavy clusters, this result has already been established, but it also
follows from one of our results. We give a general necessary condition for
nonunimodular graphs to have a phase with infinitely many heavy clusters. We
present an invariant spanning tree with on some nonunimodular graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a new way of
constructing nonunimodular graphs that have properties more peculiar than the
ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The critical Z-invariant Ising model via dimers: the periodic case
We study a large class of critical two-dimensional Ising models namely
critical Z-invariant Ising models on periodic graphs, example of which are the
classical square, triangular and honeycomb lattice at the critical temperature.
Fisher introduced a correspondence between the Ising model and the dimer model
on a decorated graph, thus setting dimer techniques as a powerful tool for
understanding the Ising model. In this paper, we give a full description of the
dimer model corresponding to the critical Z-invariant Ising model. We prove
that the dimer characteristic polynomial is equal (up to a constant) to the
critical Laplacian characteristic polynomial, and defines a Harnack curve of
genus 0. We prove an explicit expression for the free energy, and for the Gibbs
measure obtained as weak limit of Boltzmann measures.Comment: 35 pages, 8 figure
Percolation by cumulative merging and phase transition for the contact process on random graphs
Given a weighted graph, we introduce a partition of its vertex set such that
the distance between any two clusters is bounded from below by a power of the
minimum weight of both clusters. This partition is obtained by recursively
merging smaller clusters and cumulating their weights. For several classical
random weighted graphs, we show that there exists a phase transition regarding
the existence of an infinite cluster.
The motivation for introducing this partition arises from a connection with
the contact process as it roughly describes the geometry of the sets where the
process survives for a long time. We give a sufficient condition on a graph to
ensure that the contact process has a non trivial phase transition in terms of
the existence of an infinite cluster. As an application, we prove that the
contact process admits a sub-critical phase on d-dimensional random geometric
graphs and on random Delaunay triangulations. To the best of our knowledge,
these are the first examples of graphs with unbounded degrees where the
critical parameter is shown to be strictly positive.Comment: 50 pages, many figure
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