Random walk driven by two-dimensional discrete Gaussian free field

Seminar | April 26 | 3:10-4 p.m. | 1011 Evans Hall

 Marek Biskup, U.C.L.A. Mathematics

 Department of Statistics

I will discuss the random walk driven by two-dimensional pinned discrete Gaussian Free Field (pDGFF). Explicitly, I will consider the Markov chain on the square lattice that jumps across an edge with probability proportional to the exponential of the gradient of pDGFF across that edge. The chain thus tends to move in the direction of increasing values of the pDGFF and this results in trapping. I will demonstrate this effect by showing a kind of subdiffusive behavior with explicit (subdiffusive) exponents that are in agreement with conjectures from physics. The method of proof is interesting in its own right as it is based on a version of the Russo-Seymour-Welsh Theorem for effective resistance naturally associated with this problem. Based on recent joint work with Jian Ding and Subhajit Goswami.