Fluctuations in the weakly coupled 4D Anderson Hamiltonian

TL;DR

This paper investigates the behavior of the 4D Anderson Hamiltonian at weak coupling, revealing Gaussian fluctuations in the Green's function and identifying a critical coupling value where a phase transition occurs.

Contribution

It provides a rigorous analysis of Gaussian fluctuations and phase transition in the 4D Anderson model using combinatorial and renormalisation techniques.

Findings

Gaussian fluctuations characterized by an explicit variance

Identification of a critical coupling constant for phase transition

No Laplacian renormalisation needed in the model

Abstract

We study the weak coupling limit of the Anderson Hamiltonian in the critical dimension $d=4$. In a perturbative sense, we prove Gaussian fluctuations about the Green's function of the Laplacian. The fluctuations are described by an explicit effective variance, up to a critical value of the coupling constant at which we expect a phase transition in the structure of the fluctuations. The proof is based on a combinatorial analysis of Feynman diagrams, and on a detailed study of the BPHZ renormalisation of the model. We characterise the limiting distribution in terms of primitive blow-ups, and prove that no Laplacian renormalisation is present. Our approach seems applicable to a broad class of equations.