# Jacob van den Berg

## CWI, Amsterdam (The Netherlands)

- DIMACS Center
- 1st Floor Auditorium, CoRE Building
- Rutgers University
- Friday, February 28, 1997 at 11:30 am (Refreshments at 11:15 a.m.)

## Percolation, mixing and coupling

Suppose we have a finite or countably infinite graph (e.g., the
square or cubic lattice), and randomly remove edges or vertices.
What can be said about the probability that, in the resulting
(random) graph, there is a path between two given vertices,
or the probability that a given vertex belongs to an infinite
component?
Percolation theory deals with these and related questions,
and was motivated by the need for simple but clarifying models
of cooperative phenomena like the spread of infection in a
population, or a gas through a porous medium.

A random spatial process (e.g., the process describing which nodes
in a communication network are busy, or which spins in a magnetic
material are `up' or `down') is called mixing if, roughly speaking,
knowledge of the state of a given vertex gives almost no information
about the states of remote vertices. Mixing is closely related to
stability.

We will explain how percolation methods, combined with so-called
coupling arguments, are useful in the study of mixing properties
of an important class of models.

The emphasis of this talk is on intuitive ideas rather than
mathematical formalism.

dimacs-www@dimacs.rutgers.edu

Document last modified on February 13, 1997