Monotonicity in the Ising Model
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 dynamics, a Markov process whose stationary distribution is given by an Ising model.