Collaborative research: universal models of self-organized criticality

The Research Foundation of the City University of New York (RFCUNY) - College of Staten Island received a $110,000 Project Grant from the National Science Foundation's Division of Mathematical Sciences under the Mathematical and Physical Sciences program (CFDA 47.049), effective September 1, 2025 through August 31, 2026. This collaborative research award supports fundamental mathematical investigation into self-organized criticality (SOC), a phenomenon observed across natural and physical systems characterized by fractals and power-law distributions. The project aims to establish activated random walk as a rigorous mathematical model for SOC and validate theoretical predictions made by physicists regarding universal behavior in critical systems.

The primary deliverables include mathematical proofs and theoretical analysis of activated random walk models in two and higher dimensions, with specific objectives to prove the density conjecture for activated random walk, establish convergence properties of driven-dissipative versions, determine critical exponents, and analyze mixing times. The research will also extend findings to stochastic sandpile models. The award incorporates student training and educational components at the undergraduate and graduate levels, with no sub-awards planned. Results are expected to advance both mathematical understanding and physics applications of self-organized criticality theory, addressing the gap between established physicist simulations and rigorous mathematical foundations.

Abstract

Fractals and power-law tails are ubiquitous in nature and are also the hallmarks of statistical physics systems at criticality. Theoretical physicists in the late 1980S proposed that this could be explained by natural systems spontaneously converging to critical states. This theory of self-organized criticality is widely accepted by physicists based on their simulations of simple mathematical sandpile models, which appear to self-organize. But these models are poorly understood mathematically and in some cases have been proven to behave contrary to physicists' expectations. This project aims to establish that activated random walk serves as a suitable mathematical model for self-organized criticality with the universality expected by physicists, and to investigate its behavior. The results of this project will be important not just to mathematicians but to physicists as well. This project involves graduate and undergraduate students. Activated random walk and the stochastic sandpile model are two of the mathematical models thought to exhibit self-organized criticality. One concrete goal of the project is to prove the density conjecture for activated random walk in dimensions two and higher. We will do this by establishing that the driven-dissipative version of activated random walk converges to the critical density of the fixed-energy version. We will also study the mixing time of the driven-dissipative activated random walk in all dimensions and determine critical exponents of activated random walk. Finally we will establish similar results for the stochastic sandpile model. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the foundation's intellectual merit and broader impacts review criteria.- subawards are not planned for this award.