Abstract: For the site percolation model on the triangular lattice and
certain generalizations for which Cardy’s Formula has been established
we acquire a power law estimate for the rate of convergence of the
crossing probabilities to Cardy’s Formula.
We consider the time for extinction for a contact process on a tree of bounded degree as the number of vertices tends to infinity. We show that
uniformly over all such trees the extinction time tends to infinity as the
exponential of the number of vertices if the infection parameter is strictly above the critical value for the one dimensional contact process.
An application to the contact process on NSW graphs is given.
The discrete directed polymer model is a well studied example of a Gibbsian disordered system and a random walk in a random environment. The usual goal is to understand how the random environment affects the behavior of the underlying walk and how this behavior varies with a temperature parameter that determines the strength of the environment. At infinite temperature the environment has no effect and the walk is the simple random walk, while at zero temperature the environment dominates and the walk follows a single path along which the environment is largest. For temperatures in between there is a competition between the walk wanting to behave diffusively (like simple random walk) and following a path of highest energy (like last passage percolation).
In this talk I will describe recent joint work with Kostya Khanin and Jeremy Quastel for taking a scaling limit of the directed polymer model to construct a continuous path in a continuum environment. We end up with a one-parameter family of random probability measures (indexed by the temperature parameter) that we call the continuum directed random polymer. As the temperature parameter varies the paths cross over from Brownian motion to what is conjectured to be a continuum limit of last passage percolation. This cross over is an inherent feature of the KPZ universality class, which I will also briefly describe.
We derive a type of KPZ equation as a scaling limit of fluctuation fields in weakly asymmetric particle systems such as simple exclusion and zero-range processes. Joint work (in progress) with Milton Jara and Patricia Goncalves.
Recall that the notion of
generalized function is introduced for the functions
that can not be defined pointwise, and
is given as a linear functional over the test functions.
The same idea applies to random fields. In this talk,
we study the quenched asymptotics for Brownian motion
in a generalized Gaussian field. The major ingredient
includes: Solution to
an open problem posted by Carmona and Molchanov (1995) with
an answer different from what was conjectured; the quenched
laws for Brownian motions in Newtonian-type potentials, and in the potentials
driven by white noise or by fractional white noise.