# The Anderson Transition in Random Matrix Theory

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of North Carolina at Chapel Hill (NC) · $180,000

## Abstract

This project will investigate open problems at the intersection of probability theory and physics and develop new techniques for the study of the Anderson transition, a concept in condensed matter physics describing a sharp change from conducting to insulating behavior in certain materials as the density of impurities is increased. This transition plays a pivotal role in the physics of semiconductors and quantum materials. While the Anderson transition has been studied for decades, many foundational theoretical questions remain unresolved. Further, recent advances have revealed a tight connection between these questions and current research in many-body quantum systems. The project aims to provide a rigorous mathematical framework for the Anderson transition and its applications to contemporary quantum physics. The project also incorporates training of graduate students, research opportunities for undergraduate students, and the creation of new expository materials. 

A variety of mathematical models of disordered systems will be studied using tools from random matrix theory. The project will consider random matrices with heavy-tailed and sparse entries, as well as random operators on tree-like structures. The research aims to rigorously establish the existence of Anderson transitions in such models. It will also explore closely related phenomena, including multifractal eigenstates and anomalous quantum dynamics, which are believed to occur alongside Anderson transitions. T

## Key facts

- **NSF award ID:** 2450004
- **Awardee organization:** University of North Carolina at Chapel Hill (NC)
- **SAM.gov UEI:** D3LHU66KBLD5
- **PI:** Patrick Lopatto
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $180,000
- **Funds obligated:** $180,000
- **Transaction type:** Standard Grant
- **Period:** 07/01/2025 → 06/30/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2450004

## Citation

> US National Science Foundation, Award 2450004, The Anderson Transition in Random Matrix Theory. Retrieved via AI Analytics 2026-06-06 from https://api.ai-analytics.org/grant/nsf/2450004. Licensed CC0.

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