# Landscapes, games and permutations: Scaled random dynamics in statistical mechanics

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of California-Berkeley (CA) · $160,000

## Abstract

The essence of a many-body random system is often captured by its scaling limit, the structure arising when the system’s characteristic scales of space and time are brought to unit order by a linear transformation. And it is often profitable to introduce a dynamical enhancement of the system, whose snapshot at any given moment of time is the system itself, because the enhancement may have attractive properties that the original lacks, or because the behavior of the enhanced system at exceptional moments reflects aspects of the system’s scaling limit. The proposal addresses several random scaling limits and random dynamic enhancements of statistical mechanical systems. This project involves graduate student training.

Principal directions of inquiry include study of the random scaled dynamics of a random walker whose attempted linear progress is frustrated by hard obstacles that clutter the route forward;  the scale of time in a randomly evolving  energy landscape that witnesses convergence to equilibrium of the depth of the deepest valley; and the scaled counter trajectory in random-turn games played on domains in Euclidean space. The proposal will develop and exploit basic tools from the theory of Ornstein-Zernike decay and subcritical connections, and discrete harmonic analysis. By dissemination, mentorship and collaboration, the PI will seek to ensure that the research enhances the mathematical experience and trajectory of junior researchers including graduate students v

## Key facts

- **NSF award ID:** 2450726
- **Awardee organization:** University of California-Berkeley (CA)
- **SAM.gov UEI:** GS3YEVSS12N6
- **PI:** Alan Hammond
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $160,000
- **Funds obligated:** $160,000
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2026

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2450726

## Citation

> US National Science Foundation, Award 2450726, Landscapes, games and permutations: Scaled random dynamics in statistical mechanics. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2450726. Licensed CC0.

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