Gravitation versus Brownian motion

We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.

with S. Banerjee and K. Burdzy.

Lunes 9 de Mayo, 16:30 hrs. Sala 2, Facultad de matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín.

may / 2016
09

Percolation in hyperbolic space: the non-uniqueness phase

We consider Bernoulli percolation on Cayley graphs of reflection groups in the 3-dimensional hyperbolic space H^3 corresponding to a large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty no-uniqueness percolation phase, i.e., that p_c<p_u. It means that for some values of the percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph.

If time permits, I will present a sketch for the case of a right angled compact polyhedron with at least 18 faces.

Sala de seminarios del CMM, 7° piso.

may / 2016
02

Hydrodynamic limit for branching Brownian particles with spatial selection and the KPP equation.

We consider the following system in the Brunet-Derrida class: Take N

particles on the real line, each of which performs a Brownian motion

independently of any other. At rate 1/(N-1) two particles are chosen at

random, the one with the biggest position branches into two particles

and the one with the smallest position dies.

We will show that the empirical cumulative distribution associated to

this process converges in probability to the solution of the KPP

equation. Throughout the proof propagation of chaos is shown, whenever

N tends to infinity.

Additionally, for each N, we prove existence of a velocity for the

cloud of particles and a lower bound for its limit in N. Such a bound

turns out to be the minimal velocity for the KPP equation. Joint work

with Pablo Groisman and Matthieu Jonckheere.

abr / 2016
25

Some percolation processes with infinite-range dependencies and with inhomogeneous lines.

Consider the hyper-cubic lattice and remove the lines parallel to the coordinate axis independently at random. Does the set of remaining vertices undergo a sharp phase transition as the probability of removing the lines vary? How many connected components are there? In this talk we discuss these question for this model and for a continuous analogous model in which we remove cylinders from the Euclidian space in a isometry invariant way. We also discuss for Bernoulli bond percolation processes in the square lattice, how enhancing the parameter in a set of vertical lines chosen uniformly at random changes the critical point.

dic / 2015
15

Dynamic uniqueness and phase transition of chains of infinite order.

In this talk, we will introduce the notions of dynamic uniqueness

and dynamic phase transition for chain of infinite oreder. We characterize

dynamic uniqueness/phase transition by proving several equivalent

conditions. In particular, we prove that dynamic uniqueness is equivalent

to convergence in total variation distance of all the chains starting from

different pasts. We also study the relationship between our definition of

uniqueness and the $\ell^2$ criteria for the uniqueness of $g$-measures.

We prove that the Bramson-Kalikow and Hulse models exhibit dynamic

uniqueness if and only if the kernel is in $\ell^2$.

Finally, we prove that a $g$-measure $P$ is weak Bernoulli (or,

equivalently, $\beta$-mixing) if and only if $g$ exhibits dynamic

uniqueness for $P$-a.e. pasts, generalizing several results in the

literature.

Martes 24 de Noviembre, 17:00 hrs, Sala 5 facultad de matemáticas, Campus san Joaquín, PUC Chile.

nov / 2015
24

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