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Four-dimensional loop-erased random walk and uniform spanning tree
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
Venue:
Mathematical Institute, Woodstock Road OX2 6GG
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
