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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.
