About

Reza Gheissari received his PhD from New York University in 2019. After completing his doctoral studies, he held an appointment as a Miller Postdoctoral Fellow at UC Berkeley. He joined the faculty of Northwestern University in 2022. Gheissari works on probability theory and its applications. His research interests include the static and dynamic behavior of spin systems from statistical physics, as well as the relations of probability to sampling, optimization, and learning problems in high dimensions.

Research topics

• Artificial Intelligence
• Computer Science
• Algorithm
• Applied mathematics
• Geometry
• Mathematical analysis
• Mathematics
• Statistical physics
• Pure mathematics

Selected publications

• Fast relaxation of the random field Ising dynamics

  The Annals of Probability · 2026-01-01

  article
  PublisherDOI

• Mixing times of Langevin dynamics for spiked matrix models

  ArXiv.org · 2026-04-21

  articleOpen access1st authorCorresponding

  We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $θ$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $β_c(θ) = \frac{1}θ$. Namely, if $β= α/θ$, and $α<1$ then at large $θ$ the mixing time is $O(\log N)$, and if $α>1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperature $α>1$ regime circumvents the exponential bottleneck and the mixing time is $O(\log N)$. In fact, this fast mixing holds for any initialization that is symmetric with respect to the top eigenvector of the spiked matrix. Using this, we are able to show a low-temperature metastability picture, pinning down the exact exponential rate of the (worst-case initialization) mixing time for low temperatures, showing it is given by the difference of the free energies of the spiked and null models.

  PublisherOA PDF

• Mixing times of Langevin dynamics for spiked matrix models

  arXiv (Cornell University) · 2026-04-21

  preprintOpen access1st authorCorresponding

  We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $θ$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $β_c(θ) = \frac{1}θ$. Namely, if $β= α/θ$, and $α&lt;1$ then at large $θ$ the mixing time is $O(\log N)$, and if $α&gt;1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperature $α&gt;1$ regime circumvents the exponential bottleneck and the mixing time is $O(\log N)$. In fact, this fast mixing holds for any initialization that is symmetric with respect to the top eigenvector of the spiked matrix. Using this, we are able to show a low-temperature metastability picture, pinning down the exact exponential rate of the (worst-case initialization) mixing time for low temperatures, showing it is given by the difference of the free energies of the spiked and null models.

  PublisherDOI

• Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs

  ArXiv.org · 2026-04-22

  articleOpen access

  We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, approximately sampling from this model on graphs of degree at most $Δ$ is computationally hard. At parameter $p$ below the tree uniqueness threshold $p_{\mathsf{u}}(q,Δ)$, it is expected that sampling is easy and local Markov chains mix rapidly on all bounded degree graphs. On typical graphs (e.g., random regular graphs), the same is predicted at $p > p_{\mathsf{s}}(q,Δ)$, where $p_{\mathsf{s}}(q,Δ)$ is a second uniqueness transition point on the $Δ$-regular wired tree. Our first result establishes this non-uniqueness/uniqueness phase transition at $p_{\mathsf{s}}(q,Δ)$ for all $q$ on the infinite $Δ$-regular wired tree, resolving a conjecture of H{ä}ggstr{ö}m (1996). For this, we establish weak spatial mixing at $p>p_{\mathsf{s}}(q,Δ)$ under sufficiently wired boundary conditions. We use this understanding of decay of correlations to show that on the wired tree on $n$ vertices, whenever $q>1$ and $p>p_{\mathsf{s}}(q,Δ)$, the mixing time of random-cluster Glauber dynamics is a near-optimal $n^{1+o(1)}$. We then extend these results on spatial and temporal mixing from the tree to treelike geometries with mostly wired boundaries and use them to show that the random-cluster Glauber dynamics mix rapidly on the random $Δ$-regular graph for all $p>p_{\mathsf{s}}(q,Δ)$ as long as $q \ge C \log Δ$, providing an efficient sampling algorithm for both the random-cluster and Potts models in this context.

  PublisherOA PDF

• Universality for high-dimensional stochastic gradient descent

  Open MIND · 2026-01-01

  otherOpen access1st authorCorresponding

  A large family of high-dimensional statistical tasks share a common structure that their loss at a point in parameter space only depends on fixed-dimensional projections of the data (into&amp;nbsp; the&amp;nbsp;directions of the parameter and ground truth vectors). This includes mixture classification problems and single and multi-index models with one or two-layer networks. When the data distribution is isotropic Gaussian, and the parameter is trained using online stochastic gradient descent (SGD), certain low-dimensional families of “summary statistics” of the SGD trajectory evolve asymptotically autonomously, as the dimension and number of samples used go to infinity together. We consider the “universality” of these high-dimensional limits, namely whether or not&amp;nbsp;they are sensitive to the details of product data distributions beyond their first two moments. &amp;nbsp;Based on joint work with A. Jagannath.

  OA PDFDOI

• Rapid phase ordering of Ising dynamics on $\mathbb Z^2$

  arXiv (Cornell University) · 2026-05-08

  preprintOpen access1st authorCorresponding

  We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 &lt;1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p&gt;p_0$, run at any low temperature $β&gt;β_c$ converges rapidly to the plus phase measure $π^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$β$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.

  PublisherDOI

• Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs

  arXiv (Cornell University) · 2026-04-22

  preprintOpen access

  We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q&gt;2$, approximately sampling from this model on graphs of degree at most $Δ$ is computationally hard. At parameter $p$ below the tree uniqueness threshold $p_{\mathsf{u}}(q,Δ)$, it is expected that sampling is easy and local Markov chains mix rapidly on all bounded degree graphs. On typical graphs (e.g., random regular graphs), the same is predicted at $p &gt; p_{\mathsf{s}}(q,Δ)$, where $p_{\mathsf{s}}(q,Δ)$ is a second uniqueness transition point on the $Δ$-regular wired tree. Our first result establishes this non-uniqueness/uniqueness phase transition at $p_{\mathsf{s}}(q,Δ)$ for all $q$ on the infinite $Δ$-regular wired tree, resolving a conjecture of H{ä}ggstr{ö}m (1996). For this, we establish weak spatial mixing at $p&gt;p_{\mathsf{s}}(q,Δ)$ under sufficiently wired boundary conditions. We use this understanding of decay of correlations to show that on the wired tree on $n$ vertices, whenever $q&gt;1$ and $p&gt;p_{\mathsf{s}}(q,Δ)$, the mixing time of random-cluster Glauber dynamics is a near-optimal $n^{1+o(1)}$. We then extend these results on spatial and temporal mixing from the tree to treelike geometries with mostly wired boundaries and use them to show that the random-cluster Glauber dynamics mix rapidly on the random $Δ$-regular graph for all $p&gt;p_{\mathsf{s}}(q,Δ)$ as long as $q \ge C \log Δ$, providing an efficient sampling algorithm for both the random-cluster and Potts models in this context.

  PublisherDOI

• Mean-field Potts and random-cluster dynamics from high-entropy initializations

  The Annals of Applied Probability · 2026-02-01

  article

  A common obstruction to efficient sampling from high-dimensional distributions with Markov chains is the multimodality of the target distribution because they may get trapped far from stationarity. Still, one hopes that this is only a barrier to the mixing of Markov chains from worst-case initializations and can be overcome by choosing high-entropy initializations, for example, a product or weakly correlated distribution. Ideally, from such initializations, the dynamics would escape from the saddle points separating modes quickly and spread its mass between the dominant modes with the correct probabilities. In this paper, we study convergence from high-entropy initializations for the random-cluster and Potts models on the complete graph—two extensively studied high-dimensional landscapes that pose many complexities like discontinuous phase transitions and asymmetric metastable modes. We study the Chayes–Machta and Swendsen–Wang dynamics for the mean-field random-cluster model and the Glauber dynamics for the Potts model. We sharply characterize the set of product measure initializations from which these Markov chains mix rapidly, even though their mixing times from worst-case initializations are exponentially slow. Our proofs require careful approximations of projections of high-dimensional Markov chains (which are not themselves Markovian) by tractable one-dimensional random processes, followed by analysis of the latter’s escape from saddle points separating stable modes.

  PublisherDOI

• Rapid phase ordering of Ising dynamics on $\mathbb Z^2$

  ArXiv.org · 2026-05-08

  articleOpen access1st authorCorresponding

  We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 <1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p>p_0$, run at any low temperature $β>β_c$ converges rapidly to the plus phase measure $π^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$β$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.

  PublisherOA PDF

• Rapid mixing for Gibbs states within a logical sector: a dynamical view of self-correcting quantum memories

  Society for Industrial and Applied Mathematics eBooks · 2026-01-01

  book-chapter

  Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place.

  PublisherDOI

Recent grants

• Dynamics of Lattice and Mean-Field Spin Systems

  NSF · $260k · 2023–2026

Frequent coauthors

• Eyal Lubetzky

  New York University

  53 shared

• D. L. Stein

  Karlsruhe Institute of Technology

  21 shared

• Charles M. Newman

  New York University

  18 shared

• Gérard Ben Arous

  New York University

  18 shared

• Aukosh Jagannath

  University of Waterloo

  18 shared

• Antonio Blanca

  Pennsylvania State University

  13 shared

• Yuval Peres

  Beijing Institute of Mathematical Sciences and Applications

  9 shared

• Alistair Sinclair

  University of California, Berkeley

  7 shared