A ferromagnetic Ising model with periodical external field

We study the low-temperature phase diagram for a two-dimensional ferromagnetic Ising model, with a periodical external magnetic field. The external field takes two values: $h$ and $-h$. The sites associated with positive and negative values of external field mimic a configuration like a chessboard with rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero.

As a main result, we apply the reflection positivity method to show the presence of a first-order phase transition. In addition, we obtain a region of uniqueness for Gibbs measure, using a uniqueness condition obtained in terms of disagreement percolation.

Joint work with Eugene Pechersky (IITP/RAS) and Anatoly Yambartsev (IME/USP)

Sala 5, Facultad de Matemáticas, Puc Chile, 17:00 hrs.

oct / 2015
06

Metastability in a condensing zero-range process in the thermodynamic limit.

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice.

Joint work with S. Grosskinsky and M. Loulakis.

ago / 2015
25

An infinite volume dynamics for the GREM with infinitely many hierarchies

The K- process on a tree with k levels argueably represents an infinite volume dynamics for the GREM (Generalized Random Energy Model) with k hierarchies (under suitable conditions on its parameters). We take the limit of the former process as k diverges.

Lugar: Sala de Seminarios John Von Neumann, 7mo Piso CMM

Horario: 16:30.

jun / 2015
30

Optimal transport, concentration of measure and functional inequalities

This talk is devoted to Talagrand’s transport-entropy inequalities and its deep connections to the concentration of measure phenomenon, large deviation theory and logarithmic Sobolev inequalities. After an introductive part on the field, I will present recent results obtained with P-M Samson and C. Roberto characterizing different functional inequalities in terms of concentration of measure properties.

If time enables, I will also present some works in progress about transport inequalities in a discrete setting.

Lugar: Sala de Seminarios John Von Neumann, 7mo Piso CMM

Horario: 16:30.

jun / 2015
16

Two results concerning phase transitions: Sharp thresholds for van der Waerden's theorem and biased independent sets in regular graphs

Phase transitions for random structures have been of great interest in the community of discrete mathematics in the last decades. After a brief introduction into parts of this area we shall focus on two such phenomena.

First, we will consider a sharp threshold for van der Waerden’s theorem. A subset $W$ of integers satisfies the property vdW($k$), if any two coloring of $W$ yields a monochromatic arithmetic progression of length $k$.

We consider the binomial random subset of the first $n$ integers and show that as the density increases, it exhibits a swift change from not having the property vdW($k$) to having it.

Second, we discuss the hard-core distribution for finite regular graphs. Given a graph $G$ and an activation parameter $\lambda>0$, this distribution selects a random independent set $J$ of $G$ with probability proportional to $\lambda^{|J|}$. For bipartite graphs $G$ and assuming a mild spectral condition, we show that even for rather small $\lambda$ the random independent set must be essentially contained in one of the partition classes.

The talk will be in large part introductory and non-technical. The first part is based on a joint work with E. Friedgut, M. Schacht and Y. Person and the second on a joint work with P. Tetali.

jun / 2015
10

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

Beauchef 851, Edificio Norte, Piso 7,

Santiago – Chile