Counting Contours on Trees

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-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 $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift acting on $\boldsymbol{\Sigma}$, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential $A : \boldsymbol{\Sigma} \to \mathbb R$. Under certain conditions, we are able to show not only that $A$-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 $h_i = \lambda |i_d|^{-\delta}$, $\lambda$ is the wall influence, and $\delta>0$. This external field decays as it gets further away from the wall. We are able to show that when $\delta>1$ and $\beta > \beta_c(d)$, there exists a critical value $0< \lambda_c:=\lambda_c(\delta,\beta)$ such that, for $\lambda\lambda_c$ we have uniqueness of the Gibbs state. In addition, when $\delta<1$, we have only one Gibbs state for any positive $\beta$ and $\lambda$.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 $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in dimensions $d\ge 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase transition while if $\alpha<1$ there is a unique DLR state.Comment: to appear in Communications in Mathematical Physic

Entropic repulsion and lack of the gg-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 $g$-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