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Lecture notes on Quantum Diffusion and Random Matrix Theory

Felipe Hernández

TL;DR

This work studies diffusion in the weak-coupling random Schrödinger equation $i\partial_t \psi = \Delta \psi + \lambda V \psi$ and develops a two-stage, resolvent-centric framework. On the kinetic time scale, the evolution is shown to resemble free transport, with randomized scattering controlled via a non-commutative Khintchine bound and Dyson-style expansions, yielding spectral-projection and resolvent estimates. On the diffusive time scale, a local law for the Anderson model is established, together with a self-consistent equation for a key parameter and a Bethe-Salpeter-type equation for second moments, leading to diffusion-like spreading and high-probability lower bounds for resolvent mass away from the origin. The results bridge perturbative, harmonic-analysis, and random-matrix techniques to rigorously connect kinetic behavior to diffusion in disordered quantum systems, with implications for transport in random media. Key outcomes include sharp resolvent bounds, eigenfunction control, and a robust framework for energy-space diffusion via self-consistent equations and probabilistic domination.

Abstract

In joint work with Adam Black and Reuben Drogin, we develop a new approach to understanding the diffusive limit of the random Schrodinger equation based on ideas taken from random matrix theory. These lecture notes present the main ideas from this work in a self-contained and simplified presentation. The lectures were given at the summer school "PDE and Probability" at Sorbonne Université from June 16-20, 2025.

Lecture notes on Quantum Diffusion and Random Matrix Theory

TL;DR

This work studies diffusion in the weak-coupling random Schrödinger equation and develops a two-stage, resolvent-centric framework. On the kinetic time scale, the evolution is shown to resemble free transport, with randomized scattering controlled via a non-commutative Khintchine bound and Dyson-style expansions, yielding spectral-projection and resolvent estimates. On the diffusive time scale, a local law for the Anderson model is established, together with a self-consistent equation for a key parameter and a Bethe-Salpeter-type equation for second moments, leading to diffusion-like spreading and high-probability lower bounds for resolvent mass away from the origin. The results bridge perturbative, harmonic-analysis, and random-matrix techniques to rigorously connect kinetic behavior to diffusion in disordered quantum systems, with implications for transport in random media. Key outcomes include sharp resolvent bounds, eigenfunction control, and a robust framework for energy-space diffusion via self-consistent equations and probabilistic domination.

Abstract

In joint work with Adam Black and Reuben Drogin, we develop a new approach to understanding the diffusive limit of the random Schrodinger equation based on ideas taken from random matrix theory. These lecture notes present the main ideas from this work in a self-contained and simplified presentation. The lectures were given at the summer school "PDE and Probability" at Sorbonne Université from June 16-20, 2025.

Paper Structure

This paper contains 20 sections, 31 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Kinetic Time Scale
  3. The non-commutative Khintchine inequality
  4. Proof of the non-commutative Khintchine inequality
  5. Applying non-commutative Khintchine to $T_1(t)$
  6. Bounds on $T_k(t)$ from $T_1(t)$
  7. Bounds for spectral projections
  8. Resolvent bounds for $\Delta$ and $H$
  9. $\ell^p$ bounds for eigenfunctions
  10. The Diffusive Time Scale
  11. The direct perturbative approach
  12. The resolvent of a Wigner matrix
  13. More general ensembles
  14. The local law for the Anderson model
  15. The self-consistent equation for $\theta$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\psi_0$ be the Kronecker delta at the origin and $\psi_t$ solve eq:RSE on ${\mathbb Z}^d$, $d\geq 2$, and let $\kappa < \frac{1}{10}$. Then for any $\lambda^{-2}\leq T\leq \lambda^{-2-\kappa}$, the bound holds with probability at least $1 - C\lambda^{1000}$.

Figures (2)

  • Figure 1: The result of a simulation in $d=2$, $\lambda=0.1$ at time $T=2000$ with initial data $\psi_0=\mathinner{|{0}\rangle}$. On the left is the square of the wavefunction at time $T$, and on the right is the plot of $r(t)$ as a function of $t$. Note that $|\psi_T|^2$ weakly resembles a Gaussian distribution, and that $r(t)$ appears to have square-root growth at long times.
  • Figure 2: A schematic for the decomposition of the operator $T_2(t)$. The region in the blue square corresponds to a term of the form $T_1(t/2)^2$.

Theorems & Definitions (45)

  • Theorem 1.1: Simplification of Theorem 1.1 in black2025self
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4: Non-commutative Khintchine inequality
  • Corollary 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7: Matrix Jensen's inequality
  • proof
  • ...and 35 more