# Emergent universal long-range structure in random-organizing systems

## TL;DR

The paper shows that noise can lead to large-scale order in systems from physics and machine learning, revealing a universal self-organization mechanism.

## Contribution

The study uncovers a universal long-range structure across diverse systems and links it to noise correlation and hyperuniformity.

## Key findings

- Three distinct systems exhibit universal long-range behavior governed by particle noise correlation.
- Stochastic gradient descent favors flat energy landscapes, similar to hyperuniform material self-assembly.
- A fluctuating hydrodynamic theory explains the emergence of long-range order across systems.

## Abstract

Self-organization through noisy interactions is ubiquitous across physics, mathematics, and machine learning, yet how long-range structure emerges from local noisy dynamics remains poorly understood. Here, we investigate three paradigmatic random-organizing particle systems drawn from distinct domains: models from soft matter physics (random organization, biased random organization) and machine learning (stochastic gradient descent), each characterized by distinct sources of noise. We discover universal long-range behavior across all systems, namely the suppression of long-range density fluctuations, governed solely by the noise correlation between particles. Furthermore, we establish a connection between the emergence of long-range structure and the tendency of stochastic gradient descent to favor flat regions of energy landscape—a phenomenon widely observed in machine learning. To rationalize these findings, we develop a fluctuating hydrodynamic theory that quantitatively captures all observations. Our study resolves long-standing questions about the microscopic origin of noise-induced hyperuniformity, uncovers striking parallels between stochastic gradient descent dynamics on particle system energy landscapes and neural network loss landscapes, and should have wide-ranging applications—from the self-assembly of hyperuniform materials to ecological population dynamics and the design of generalizable learning algorithms.

Noise is usually associated with disorder, but it can also generate large-scale order. Here, the authors show that three distinct systems, spanning soft matter and stochastic optimization, self-organize into the same universal long-range structure, pointing to a shared self-organization mechanism.