# DIMACS Discrete Mathematics Seminar

## Title:

Generating random surfaces, tilings and eulerian orientations

## Speaker:

- Dana Randall
- Institute for Advanced Study and DIMACS

## Place:

- Seminar Room 431, CoRE Building,
- Busch Campus, Rutgers University.

## Time:

- 4:30 PM
- Tuesday, February 14, 1995

## Abstract:

We consider the problems of efficiently generating random dimer
configurations (or tilings) and random Eulerian orientations on
a finite region of a two-dimensional lattice. Thurston, Conway
and Lagarias showed that given arbitrary boundary conditions on
a region of the lattice, the set of dimer coverings map bijectively
to a set of piecewise linear surfaces with a fixed boundary.
Similarly, the set of Eulerian orientations, or configurations
of the ice model in statistical mechanics, are known to correspond
bijectively with ``solid-on-solid'' surfaces where the difference in
height between any two neighboring points is exactly one. We
define a natural Markov chain for each family of surfaces and show
that each will converge quickly to the uniform distribution over
configurations. The proofs of the convergence rates rely on
simple coupling arguments. (This is joint work with M. Luby and
A. Sinclair.)

Document last modified on February 9, 1995