The four-dimensional Anderson model: a case study for critical SPDEs

We study the weakly coupled elliptic Anderson model with spatial white noise on the four-dimensional torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With coupling $\lambda |\log\varepsilon|^{-\frac12}$ where $\lambda>0$ is sufficiently small, we prove that the Green's function of the corresponding random Schr\"odinger 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 $|\log\varepsilon|$, while the perturbative expansion contains factorially many pairings and a growing number of renormalization terms. To overcome this, we construct a truncated renormalized parametrix and prove sharp high-order bounds for its remainder. A central ingredient is 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.