About

Christopher Hoffman is a professor in the Department of Mathematics at the University of Washington in Seattle. His work includes research on stable matchings in the plane, as evidenced by a collaborative project with Ander Holroyd and Yuval Peres. Further details about his research focus, background, or key contributions are not provided in the available page text.

Research topics

• Mathematics
• Physics
• Quantum mechanics
• Statistical physics
• Combinatorics
• Computer Science
• Pure mathematics
• Mathematical physics
• Sociology
• Mathematical analysis
• Philosophy
• Demography
• Statistics
• Geometry

Selected publications

• Activated random walk exhibits self-organized criticality

  arXiv (Cornell University) · 2026-04-30

  articleOpen access1st authorCorresponding

  To explain the ubiquity of power laws and fractals in nature, Bak, Tang, and Wiesenfeld formulated simple conditions for a system to self-organize into a critical state. Dickman, Muñoz, Vespignani, and Zapperi postulated that the self-organized critical state matches the critical state in corresponding fixed-energy models undergoing traditional phase transitions. Although the theory has been applied broadly over the past five decades, no mathematical model has been proven to exhibit the conjectured behavior. Indeed, the originally proposed abelian sandpile model displays nonuniversal behavior stemming from its slow mixing. Marking the first result of its kind, we prove that the 1-d activated random walk model mixes quickly into a stationary state with power-law avalanches and limiting critical density that equals the critical value for the fixed-energy version.

  PublisherOA PDF

• The hockey-stick conjecture for activated random walk

  Proceedings of the American Mathematical Society · 2026-02-02

  article1st authorCorresponding
  PublisherDOI

• Explosivity in 1-d Activated Random Walk

  HAL (Le Centre pour la Communication Scientifique Directe) · 2026-01-06

  preprintOpen access

  We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$.

  PublisherDOI

• Local Density of Activated Random Walk on $\mathbb{Z}$

  arXiv (Cornell University) · 2026-01-12

  preprintOpen access1st authorCorresponding

  We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.

  PublisherDOI

• Local Density of Activated Random Walk on $\mathbb{Z}$

  ArXiv.org · 2026-01-12

  articleOpen access1st authorCorresponding

  We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.

  PublisherOA PDF

• Explosivity in 1-d Activated Random Walk

  ArXiv.org · 2026-01-06

  articleOpen access

  We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$.

  PublisherOA PDF

• Activated random walk exhibits self-organized criticality

  arXiv (Cornell University) · 2026-04-30

  preprintOpen access1st authorCorresponding

  To explain the ubiquity of power laws and fractals in nature, Bak, Tang, and Wiesenfeld formulated simple conditions for a system to self-organize into a critical state. Dickman, Muñoz, Vespignani, and Zapperi postulated that the self-organized critical state matches the critical state in corresponding fixed-energy models undergoing traditional phase transitions. Although the theory has been applied broadly over the past five decades, no mathematical model has been proven to exhibit the conjectured behavior. Indeed, the originally proposed abelian sandpile model displays nonuniversal behavior stemming from its slow mixing. Marking the first result of its kind, we prove that the 1-d activated random walk model mixes quickly into a stationary state with power-law avalanches and limiting critical density that equals the critical value for the fixed-energy version.

  PublisherDOI

• Cutoff for activated random walk

  ArXiv.org · 2025-01-29

  preprintOpen access1st authorCorresponding

  We prove that the mixing time of driven-dissipative activated random walk on an interval of length $n$ with uniform or central driving exhibits cutoff at $n$ times the critical density for activated random walk on the integers. The proof uses a new result for arbitrary graphs showing that the chain is mixed once activity is likely at every site.

  PublisherOA PDFDOI

• The hockey-stick conjecture for activated random walk

  arXiv (Cornell University) · 2024-11-04

  preprintOpen access1st authorCorresponding

  We prove a conjecture of Levine and Silvestri that the driven-dissipative activated random walk model on an interval drives itself directly to and then sustains a critical density. This marks the first rigorous confirmation of a sandpile model behaving as in Bak, Tang, and Wiesenfeld's original vision of self-organized criticality.

  PublisherOA PDFDOI

• Activated Random Walks on $\mathbb{Z}$ with Critical Particle Density

  arXiv (Cornell University) · 2024-11-12

  preprintOpen access

  The Activated Random Walk (ARW) model is a promising candidate for demonstrating self-organized criticality due to its potential for universality. Recent studies have shown that the ARW model exhibits a well-defined critical density in one dimension, supporting its universality. In this paper, we extend these results by demonstrating that the ARW model on $\mathbb{Z}$, with a single initially active particle and all other particles sleeping, maintains the same critical density. Our findings relax the previous assumption that required all particles to be initially active. This provides further evidence of the ARW model's robustness and universality in depicting self-organized criticality.

  PublisherOA PDFDOI

Recent grants

• Plaquette Percolation

  NSF · $285k · 2013–2017

• Probability Postdoctoral Training Center

  NSF · $550k · 2015–2022

• Planar First Passage Percolation

  NSF · $270k · 2017–2022

• Invariant Measures for Random Growth Processes

  NSF · $220k · 2008–2012

Frequent coauthors

• Shirshendu Ganguly

  University of California, Berkeley

  22 shared

• Elisabetta Candellero

  Roma Tre University

  18 shared

• Lionel Levine

  18 shared

• Riddhipratim Basu

  International Centre for Theoretical Sciences

  14 shared

• Yuval Peres

  Beijing Institute of Mathematical Sciences and Applications

  12 shared

• Matthew Kahle

  11 shared

• Douglas Rizzolo

  11 shared

• Alexander E. Holroyd

  University of Bristol

  11 shared

Labs

• Math AI LabPI

Awards & honors

• 2016 Simons Fellow
• AMS Centennial Fellowship (2008)