Noise sensitivity for Voronoi Percolation

Noise sensitivity is a concept that measures if the outcome of Boolean function can be predicted when one is given its value for a perturbation of the configuration. A sequence of functions is noise sensitive when this is asymptotically not possible. A non-trivial example of a sequence that is noise sensitive is the crossing functions in critical two-dimensional Bernoulli bond percolation. In this setting, noise sensitivity can be understood via the study of randomized algorithms. Together with a discretization argument, these techniques can be extended to the continuum setting. In this talk, we prove noise sensitivity for critical Voronoi percolation in dimension two, and derive some consequences of it.

Based on a joint work with D. Ahlberg.

May / 2018
03

Harnack inequality for degenerate balanced random walks.

We consider an i.i.d. balanced environment $\omega(x,e)=\omega(x,-e)$,genuinely d dimensional on the lattice and show that there exist a

positive constant $C$ and a random radius $R(\omega)$ with streched

exponential tail such that every non negative

$\omega$ harmonic function $u$ on the ball $B_{2r}$ of radius

$2r>R(\omega)$,

we have $\max_{B_r} u <= C \min_{B_r} u$. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow’s Harnack’s inequality for i.i.d. bond percolation to the directed case.

This is joint work with N.Berger M. Cohen and X. Guo.

Sala 3, Facultad de Matemáticas, PUC

Apr / 2018
02

“Invariant measures of discrete interacting particle systems: Algebraic aspects”

We consider a continuous time particle system on a graph L being either Z, Z_n, a segment {1,…, n}, or Z^d, with state space Ek={0,…,k-1} for some k belonging to {infinity, 2, 3, …}. We also assume that the Markovian evolution is driven by some translation invariant local dynamics with bounded width dependence, encoded by a rate matrix T. These are standard settings, satisfied by many studied particle systems. We provide some sufficient and/or necessary conditions on the matrix T, so that this Markov process admits some simple invariant distribution, as a product measure, as the distribution of a Markov process indexed by Z or {1,…, n} (if L=Z or {1,…,n}), or as a Gibbs measure (if L=Z_n). These results are mainly obtained with some manipulations of finite words, with alphabet Ek, representing subconfigurations of the systems. For the case L=Z, we give a procedure to find the set of invariant i.i.d and Markov measures.

Sala John Von Neumann, 7mo piso, CMM, 15:00 hrs.

Jan / 2018
09

Construction of geometric rough paths

This talk is based on a joint work in progress with L. Zambotti (UPMC). First, I will give a brief introduction to the theory of rough paths focusing on the case of Hölder regularity between 1/3 and 1/2. After this, I will address the basic problem of construction of a geometric rough path over a given ɑ-Hölder path in a finite-dimensional vector space. Although this problem was already solved by Lyons and Victoir in 2007, their method relies on the axiom of choice and thus is not explicit; in exchange the proof is simpler. In an upcoming paper, we provide an explicit construction clarifying the connection between rough paths theory and free (nilpotent) Lie algebras. In particular, we use an explicit form of the Baker–Campbell–Hausdorff formula due to Loday in order to provide explicit expressions and bounds to achieve such a construction.

Sala de seminarios del 5to piso, CMM, 16:00 hrs.

Nov / 2017
27

From Anderson models to GOE statistics

We first prove some SDE limit for product of random matrices. We then apply this to transfer matrices of block-Jacobi operators which we use to obtain limiting statistics for Anderson models on long strips under proper rescaling of the randomness. With the correct sequence of limits we obtain a random matrix ensemble and finally the Sine_1 kernel.

Finally we construct a sequence of graphs (antitrees) where some averaging effect of a random potential mimics the rescaling in the step before. This way we obtain a sequence of random matrices with randomness of fixed strength (disorder) only along the diagonal for which we have limiting GOE statistics (Sine_1 kernel).

Sala 5, Facultad de Matemáticas, PUC. 17:00 hrs.

Nov / 2017
20

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