Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been established for Ising and \Phi^4 models for d \geq 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\geq 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.

**Date**: 14 May 2018, 12:00 (Monday, 4th week, Trinity 2018)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Speaker**: Wei Wu (Department of Statistics, University of Warwick)**Organising department**: Department of Statistics**Organisers**: Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Public- Editors: Christina Goldschmidt, James Martin