Session: Probability, Combinatorics, and Statistical Mechanics<br>Organizer:
Friday, August 5, 2005 - 8:45am - 10:30am
Russell Lyons (Indiana University)
- Simple random surfaces
Richard Kenyon (University of British Columbia)
This is joint work with David Brydges and Jessica Young. We study a model of random surfaces coming with an immersion into an arbitrary two-complex. Certain probabilistic quantities can be computed using the Green's function for the Laplacian on 1-forms.
- Infinite volume limit of the Abelian sandpile model on Zd
Antal Jarai (Carleton University)
The Abelian 'sandpile' model was introduced by physicists as a basic example of self-organized criticality (SOC). Roughly speaking, SOC arises when a stochastic dynamics drives a system towards a stationary state characterized by power laws. We study existence of the infinite volume limit for the model, and properties of this limit, giving insight into the asymptotic behaviour of large 'sandpiles'. Most progress can be made above the upper critical dimension d > 4. We discuss some open problems related to extending these results to lower dimensions.
- Tug of war and the infinity Laplacian
Scott Sheffield (New York University)
The infinity Laplacian (informally, the second derivative in the gradient direction) is a simple yet mysterious operator with many applications.
Tug of war is a two player random turn game played as follows:
SETUP: Assign each player one of two disjoint target sets T_1 and T_2 in the plane, and fix a starting position x and a constant epsilon. Place the game token at x.
GAME PLAY: Toss a fair coin and allow the player who wins the coin toss to move the game token up to epsilon units in the direction of his or her choice. Repeat the above until the token reaches a target set T_i. The ith player is then declared the winner.
Given parameters epsilon and x, write u_epsilon(x) for the probability that player one wins when both players play optimally. We show that as epsilon tends to zero, the functions u_epsilon(x) converge to the infinity harmonic function with boundary conditions 1 on T_1 and 0 on T_2.
Our strategic analysis of tug of war leads to new formulations and significant generalizations of several classical results about infinity laplacians. The game theoretic arguments are simpler and more elementary than the original proofs.
This talk is based on joint work with Yuval Peres, Oded Schramm, and David Wilson.