##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability Seminar

## Wei Wu

#### NYU

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

##### Abstract:

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 $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ 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, by showing 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.

Host: Bruce Driver

### October 20, 2016

### 10:00 AM

### AP&M 6402

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