30 research outputs found
Counting Contours on Trees
We calculate the exact number of contours of size containing a fixed
vertex in -ary trees and provide sharp estimates for this number for more
general trees. We also obtain a characterization of the locally finite trees
with infinitely many contours of the same size containing a fixed vertex.Comment: 12 pages, 2 figure
Weak KAM methods and ergodic optimal problems for countable Markov shifts
Let be the left shift
acting on , a one-sided Markov subshift on a countable
alphabet. Our intention is to guarantee the existence of -invariant
Borel probabilities that maximize the integral of a given locally H\"older
continuous potential . Under certain
conditions, we are able to show not only that -maximizing probabilities do
exist, but also that they are characterized by the fact their support lies
actually in a particular Markov subshift on a finite alphabet. To that end, we
make use of objects dual to maximizing measures, the so-called sub-actions
(concept analogous to subsolutions of the Hamilton-Jacobi equation), and
specially the calibrated sub-actions (notion similar to weak KAM solutions).Comment: 15 pages. To appear in Bulletin of the Brazilian Mathematical
Society
Phase Transitions in the semi-infinite Ising model with a decaying field
We study the semi-infinite Ising model with an external field , is the wall influence, and . This
external field decays as it gets further away from the wall. We are able to
show that when and , there exists a critical
value such that, for
we
have uniqueness of the Gibbs state. In addition, when , we have only
one Gibbs state for any positive and .Comment: 16 pages, 6 figure
Phase Transitions in Ferromagnetic Ising Models with spatially dependent magnetic fields
In this paper we study the nearest neighbor Ising model with ferromagnetic
interactions in the presence of a space dependent magnetic field which vanishes
as , , as . We prove that in
dimensions for all large enough if there is a phase
transition while if there is a unique DLR state.Comment: to appear in Communications in Mathematical Physic
Entropic repulsion and lack of the -measure property for Dyson models
We consider Dyson models, Ising models with slow polynomial decay, at low
temperature and show that its Gibbs measures deep in the phase transition
region are not -measures. The main ingredient in the proof is the occurrence
of an entropic repulsion effect, which follows from the mesoscopic stability of
a (single-point) interface for these long-range models in the phase transition
region.Comment: 22 pages, 4 figure