Loop erased random walk, uniform spanning tree, and bi-Laplacian Gaussian field in the critical dimension

Wednesday, September 21, 2016 -
3:00pm to 4:00pm
Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and Phi^4 models for d \ge 4. We describe a simple spin model from uniform spanning forests in Z^d whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for d\ge 4. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4: we show that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.
Speaker: 
Wei Wu
Courant Institute/NYU
Event Location: 
Fine Hall 214