Department of Mathematics

Colloquium

  •  Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.
  •  12/01/2016
  •  4:10 PM - 5:00 PM
  •  C304 Wells Hall
  •  Wei Wu, Courant

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 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, 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.

 

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