42 research outputs found
The survival of large dimensional threshold contact processes
We study the threshold contact process on with
infection parameter . We show that the critical point
, defined as the threshold for survival starting from
every site occupied, vanishes as . This implies that the threshold
voter model on has a nondegenerate extremal invariant
measure, when is large.Comment: Published in at http://dx.doi.org/10.1214/08-AOP440 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dobrushin-Kotecký-Shlosman theorem up to the critical temperature
We develop a non-perturbative version of the Dobrushin-Koteck'y-Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical
Metastable behavior for bootstrap percolation on regular trees
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law
given by Bernoulli(p). The sites are updated according to the usual rule: a
vacant site becomes occupied if it has at least theta occupied neighbors,
occupied sites remain occupied forever. It is known that, when b>theta>1, the
limiting density q=q(p) of occupied sites exhibits a jump at some
p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We
investigate the metastable behavior associated with this transition.
Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the
system lingers around the "critical" state for time order h^{-1/2} and then
passes to fully occupied state in time O(1). The law of the entire
configuration observed when the occupation density is q in (q_t,1) converges,
as h tends to 0, to a well-defined measure.Comment: 10 pages, version to appear in J. Statist. Phy
Facilitated oriented spin models:some non equilibrium results
We analyze the relaxation to equilibrium for kinetically constrained spin
models (KCSM) when the initial distribution is different from the
reversible one, . This setting has been intensively studied in the physics
literature to analyze the slow dynamics which follows a sudden quench from the
liquid to the glass phase. We concentrate on two basic oriented KCSM: the East
model on \bbZ, for which the constraint requires that the East neighbor of
the to-be-update vertex is vacant and the model on the binary tree introduced
in \cite{Aldous:2002p1074}, for which the constraint requires the two children
to be vacant. While the former model is ergodic at any , the latter
displays an ergodicity breaking transition at . For the East we prove
exponential convergence to equilibrium with rate depending on the spectral gap
if is concentrated on any configuration which does not contain a forever
blocked site or if is a Bernoulli() product measure for any . For the model on the binary tree we prove similar results in the regime
and under the (plausible) assumption that the spectral gap is
positive for . By constructing a proper test function we also prove that
if and convergence to equilibrium cannot occur for all
local functions. Finally we present a very simple argument (different from the
one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results
and ``energy barrier'' considerations, which yields the sharp upper bound for
the spectral gap of East when .Comment: 16 page