# Bay Area Discrete Math Day XII: Monotonicity in the Ising Model

Google TechTalks

Bay Area Discrete Math Day XII

April 15, 2006

Nicholas Weininger (Google Inc.)

ABSTRACT

The Ising model assigns probabilities to "spin configurations" of the vertices of a graph G in which each vertex is given spin +1 or -1. In a ferromagnetic Ising model, for any v, w ?V(G), setting the spin of v to +1 makes the spin at w more likely to be +1. A natural and elegant conjecture due to Kenyon, Mossel, and Peres states that the strength of this "influence" of v on w can only decrease when some other vertices' spins are fixed. We prove some partial results in this direction and discuss the obstacles to a full proof. We also discuss a related conjecture of Peres on a monotonicity property of the heat bath dynamic, a Markov process whose stationary distribution is given by an Ising model. «