Corner percolation and the square root of 17

January 24, 2006
Gabor Pete | University of Berkeley

We consider a dependent bond percolation model on Z², introduced by Balint Toth, in which every edge is present with probability 1/2, and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. A more detailed analysis leads to the derivation of the following critical exponents: the tail probability Pr (diameter of the cycle of the origin > n) ≅ n^-γ, and the expectation E (length of a cycle conditioned on having diameter n) ≅ n^δ. We show that γ=(5-√{17})/4=0.219… and δ=(√{17}+1)/4=1.28… The relation γ+δ=3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the Additive Brownian Motion, whose level sets have Hausdorff dimension 3/2. The value of δ comes from the solution of a singular sixth order ODE.

Speaker Details

I was born in 1976 in Hungary and did my undergrad studies there, at the Univ of Szeged. My Master’s thesis was based on a probabilistic combinatorics paper published in Random Struct and Algorithms (with J Balogh). I also spent a year in Cambridge, UK, the Part III course, to learn modern geometry. I wrote my essay on E Witten’s supersymmetric approach to Morse theory. I started my PhD in Berkeley, under the supervision of Yuval Peres, in 2001. My research interests have been in critical random systems and their scaling limits; connections between stoch processes (random walk, percolation) and the large-scale geometry of the underlying graph; and a probabilistic game theory approach to understand certain PDE’s.