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.

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.