# Seminarios

## Leandro Pimentel

### Título:

Duality between coalescence times and exit points in last-passage percolation models.

### Abstract:

In this talk we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy process.

Location: TBA

may / 2015 20

## Enrique Andjel

### Título:

First passage percolation and escape strategies

### Abstract:

Abstract: Consider first passage percolation on ${Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_\pi(u,v)$ be the time from $u$ to $v$ for a path $\pi$ and $t(u,v)$ the minimal time among all paths from $u$ to $v$. We ask whether or not there exist points $x,y \in {Z}^d$ and a semi-infinite path $\pi=(y_0=y,y_1,\dots)$ such that \$t_\pi(y, y_{n+1})

mar / 2015 10 - 12

## Rodrigo Bissacot

### Título:

Phase Transitions in Ferromagnetic Ising Models with magnetic fields

### Abstract:

We study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field of type 1/|x|^{α}. We prove that in dimensions d ≥ 2 for all β large enough if α > 1 there is a phase transition while if α < 1 there is a unique DLR state. Jointly with Marzio Cassandro (GSSI, L’Aquila), Errico Presutti (GSSI, L’Aquila) and Leandro Cioletti (Unb, Brazil). The paper is available in arxiv.

Location: PUC, Math Department, Sala 1

abr / 2014 22

## Manuel Cabezas

### Título:

The Alexander-Orbach conjecture on regular trees.

### Abstract:

The Alexander Orbach conjecture states that the Incipient Infinite Cluster of percolation has spectral dimension 4/3. In this talk we give a brief overview of the historic development of the conjecture and present some new related results. More specifically, we provide an scaling limit fot the random walk on the incipient infinite cluser of a regular tree projected to the backbone.

Location: USACH, Edificio CITECAMP, sala de seminario

abr / 2014 15

## Joaquín Fontbona

### Título:

Quantitative rates of convergence for the renewal theorem with spread-out distributions.

### Abstract:

We present some partial results on quantitative exponential convergence estimates for the renewal theorem when the inter arrival distribution has a uniform component and one finite exponential monent. We improve to that end the classic idea of coupling the age processes of a delayed renewal process and a non delayed one in order to get explicit estimates. A key element is our construcion of a coalescent coupling between two copies of the process that succeeds with at least some explicit probability which is independent of their initial relative delay. This construction and the computation of the lower bound rely on the use of the uniform component to couple inter arrival lengths with small enough “shifted” of those random variables. A second element is the study of a biased random walk associated with two independent copies of the process, which allows us to obtain, via Lyapunov and martingale techniques, an estimate of the total elapsed time required to observe renewals of the two copies within a time-interval of (large enough) prescribed length. Based on joint work in progress with Jean-Baptiste Bardet and Alejandra Christen.

abr / 2014 01
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##### Departamento de Matemáticas

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

##### Centro de Modelamiento Matemático (CMM)

Facultad de Ciencias Físicas y Matemáticas (FCFM)