Self-consistent equations and quantum diffusion for the Anderson model

2025/06/06 by Black, Adam, Drogin, Reuben, Hernández, Felipe · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.06468

Abstract

We consider the Anderson tight-binding model on ℤ^d, d≥ 2, with Gaussian noise and at low disorder λ>0. We derive a diffusive scaling limit for the entries of the resolvent R(z) at imaginary part \operatorname*Im z∼λ^2+κ_d, κ_d>0, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for d=2 are the first quantum diffusion results for the Anderson model on ℤ². The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for R(z). This is facilitated by new estimates for ‖R(z)‖_{ℓ^p→ ℓ^q} that control the recollisions.

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Cited by

• Lecture notes on Quantum Diffusion and Random Matrix Theory

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