MARK KAC SEMINAR

We prove rigorous inequalities for some of the critical exponents of the Abelian sandpile model on Z^d, d≥2. The exponents we consider are for the toppling probability, the avalanche radius and the avalanche size. The first part of the talk will cover background about the model. The second part is based on joint work with S. Bhupatiraju and J. Hanson.

A configuration of the planar random current model can be viewed as a collection of dual Ising contours together with an independent Bernoulli bond percolation with prescribed success probabilities. The double random current model is simply a superimposition of two iid random current configurations. Its clusters are composed of XOR-Ising contours and of additional components arising from the percolation process or two overlapping single Ising contours. For each such cluster C we toss an independent ±1 symmetric coin X_C. A cluster C is called odd around a face u if the contours contained in C assign spin −1 to u under +1 boundary conditions. The double random current nesting field at u is defined to be the sum of X_C over clusters C odd around u.

I will provide a measure-preserving mapping between double currents and dimers on a particular bipartite graph. Under this map the nesting field becomes the height function of the dimer model. Using this connection together with the results of Kenyon, Okounkov and Sheffield on the dimer model, I will prove that the magnetization of the critical Ising model on any biperiodic graph vanishes. This is partially joint work with Hugo Duminil-Copin.