Limiting laws for some integrated processes

The study of limiting laws, or penalizations, of a given process may

be seen (in some sense) as a way to condition a probability law by an

a.s. infinite random variable. The systematic study of such problems

started in 2006 with a series of papers by Roynette, Vallois and Yor

who looked at Brownian motion perturbed by several examples of

functionals. These works were then generalized to many families of

processes: random walks, Lévy processes, linear diffusions…

We shall present here some examples of penalization of a non-Markov

process, i.e. the integrated Brownian motion, by its first passage

time, nth passage time, and last passage time up to a finite horizon. We shall show that the penalization principle holds in all these cases, but that the

conditioned process does not always behave as expected. Recent results

around persistence of integrated symmetric stable processes will also

be discussed.

Aug / 2016
11

Martingale problem and trace processes applied to metastability.

The Martingale problem is a concept introduced by Stroock and Varadhan

which can be understood as a sort of ordinary differential equation in

which the vector field is replaced by a field of second order

differential operators. A Markov process can be characterized as a

unique solution of a Martingale problem. This fact turns the

martingale problem in a very useful tool to prove convergence of

stochastic processes derived from Markov processes.

In this talk we shall use the martingale problem to prove the

convergence of processes arising in the study of metastable systems.

We shall explain how this tool is used in combination with other ones

like trace processes and potential theory. Finally, we shall show some

examples of systems in which this approach has been applied.

This is a joint work with C. Landim.

Aug / 2016
01

Phase Transitions on the Long Range Ising Models in presence of an random external field

We study the ferromagnetic one-dimensiosnal Random Field Ising Model with (RFIM) in presence of an external random field. The interaction between two spins decays as $d^{\alpha-2}$ where $d$ is the distance between two sites and $\alpha \in [0,1/2)$ is a parameter of the model. We consider an external random field on $\mathbb{Z}$ with independent but not identically distributed random variables. Specifically for each $i \in \mathbb{Z}$, the distribution of $h_i$ is

$P[h_i=\pm \theta(1+|i|)^{-\nu/2}]$,

This work, whose main goal is the study of the existence of a phase transition at a strictly positive temperature for different values of $\nu$ is inspired on the very recent article [2] where the 2D Ising Model with spatially dependent but not random external field is studied. In the random case, we combine some of the martingale difference techniques used in the previous articles of Cassandro, Picco and Orlandi [3], and the Aizemann & Wehr method [3]. Some of the classical results, the key parts of this work and some of the technical difficulties will be discussed in this talk.

Joint work with Pierre Picco

References:

[1] M. Aizenman and C. M. Newman. Discontinuity of the percolation density in one-dimensional 1/|x − y|

107(4):611–647, 1986.

[2] Rodrigo Bissacot, Marzio Cassandro, Leandro Cioletti, and Errico Presutti.

Phase transitions in ferromagnetic ising models with spatially dependent

magnetic fields. Communications in Mathematical Physics, 337(1):41–53,

2015.

[3] Marzio Cassandro, Enza Orlandi, and Pierre Picco. Phase transition in the

1d random field Ising model with long range interaction. Communications

in Mathematical Physics, 288(2):731–744, 2009.

Jul / 2016
25

The Vertex Reinforced Jump Process and a Random Schrödinger Operator.

We will give an overview on recent developpements on two self-interacting processes : the Edge Reinforced Random Walk (ERRW)

and the Vertex Reinforced Jump Process (VRJP).

The ERRW and VRJP are known since a few years to be related to a supersymetric field

considered by Disertori, Spencer and Zirnbauer.

On finite graphs we introduce a random Schrödinger operator with a

random potential, with decorrelation at distance 2, from which the

mixing field of the VRJP can be described from the Green function.

The distribution of this potential is explicite and appears to be new,

and can be understood as a multivariate generalization of the inverse

gaussian law.

Interesting phenomenons appear by extending this representation to

infinite graphs.

In particular, the transience of the VRJP is signed by the existence of

a positive generalized eigenfunction of the random Schrödinger operator.

The VRJP can then be represented as a mixture of Markov Jump processes

with a field involving the Green function of the random Schrödinger

operator, the generalized eigenfunction and an extra independent random

variable that governs the escape probability.

From this we can infer a functional CLT at weak disorder for the VRJP

and ERRW in dimension d>2.

Based on joint works with, P. Tarrès, X. Zeng.

Jul / 2016
21

The log-Sobolev inequality for unbounded spin systems on the lattice.

A criterion will be presented for the log-Sobolev inequality for unbounded

spin systems on the lattice with non-quadratic interactions. This is a

joint work with Takis Konstantopoulos (Uppsala) and James Inglis (INRIA).

Furthermore, in the case of quadratic interactions, a perturbation result

for the inequality will be presented.

Jun / 2016
13

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