Hao Shen (U Chicago). The 4D Anderson model: a case study for critical SPDEs

We study the weakly coupled elliptic Anderson model with spatial white noise on the 4D torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With sufficiently small coupling, we prove that the Green's function of the corresponding random Schrödinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance. The main difficulty is that, for such critical models, one must expand up to order |logε|, while the perturbative expansion contains factorially  many pairings and a growing number of renormalization terms.  To overcome this, we develop a multiscale analysis based on a new version  of Hepp trees, combined with new estimates for summations over  permutations. These estimates reveal a precise balance between  logarithmic losses from scale summation and factorial gains  from the structure of primitive pairings.  The methods developed here are intended as a first step toward a general theory for critical SPDEs with weak couplings.  Based on joint work with Yu Deng.