arXiv subjects

Reza Gheissari

Publications and source records attributed to Reza Gheissari.

At least 19 recordsLinked to original sources

arXiv2026

On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces

We consider integer-valued $\nablaϕ$ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions $d\ge3$, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.

math.PR↗

arXiv2026

Mixing of general biased adjacent transposition chains

We analyze the general biased adjacent transposition shuffle process, which is a well-studied Markov chain on the symmetric group $S_n$. In each step, an adjacent pair of elements $i$ and $j$ are chosen, and then $i$ is placed ahead of $j$ with probability $p_{ij}$. This Markov chain arises in the study of self-organizing lists in theoretical computer science, and has close connections to exclusion processes from statistical physics and probability theory. Fill (2003) conjectured that for general $p_{ij}$ satisfying $p_{ij} \ge 1/2$ for all $i 0$, as long as $p_{ij} >1/2+\varepsilon$ for all $i<j$, the mixing time is $Θ(n^2)$ and exhibits pre-cutoff. Our key technical result is a form of spatial mixing for the general biased transposition chain after a suitable burn-in period. In order to use this for a mixing time bound, we adapt multiscale arguments for mixing times from the setting of spin systems to the symmetric group.

math.PR↗

arXiv2026

Mixing times of Langevin dynamics for spiked matrix models

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.

math.PR↗

arXiv2026

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

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.

math.PR↗

arXiv2026

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

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.

math.PR↗

arXiv2026

Local geometry of high-dimensional mixture models: Effective spectral theory and dynamical transitions

We study the local geometry of empirical risks in high dimensions via the spectral theory of their Hessian and information matrices. We focus on settings where the data, $(Y_\ell)_{\ell =1}^n \in \mathbb{R}^d$, are i.i.d. draws of a $k$-Gaussian mixture model, and the loss depends on the projection of the data into a fixed number of vectors, namely $\mathbf{x}^\top Y$, where $\mathbf{x}\in \mathbb{R}^{d\times C}$ are the parameters, and $C$ need not equal $k$. This setting captures a broad class of problems such as classification by one and two-layer networks and regression on multi-index models. We provide exact formulas for the limits of the empirical spectral distribution and outlier eigenvalues and eigenvectors of such matrices in the proportional asymptotics limit, where the number of samples and dimension $n,d\to\infty$ and $n/d=ϕ\in (0,\infty)$. These limits depend on the parameters $\mathbf{x}$ only through the summary statistic of the $(C+k)\times (C+k)$ Gram matrix of the parameters and class means, $\mathbf{G} = (\mathbf{x},\boldsymbolμ)^\top(\mathbf{x},\boldsymbolμ)$. It is known that under general conditions, when $\mathbf{x}$ is trained by online stochastic gradient descent, the evolution of these same summary statistics along training converges to the solution of an autonomous system of ODEs, called the effective dynamics. This enables us to connect the training dynamics to the spectral theory of these matrices generated with test data. We demonstrate our general results by analyzing the effective spectrum along the effective dynamics in the case of multi-class logistic regression. In this setting, the empirical Hessian and information matrices have substantially different spectra, each with their own static and even dynamical spectral transitions.

math.ST↗

arXiv2025

Universality of high-dimensional scaling limits of stochastic gradient descent

We consider statistical tasks in high dimensions whose loss depends on the data only through its projection into a fixed-dimensional subspace spanned by the parameter vectors and certain ground truth vectors. This includes classifying mixture distributions with cross-entropy loss with one and two-layer networks, and learning single and multi-index models with one and two-layer networks. When the data is drawn from an isotropic Gaussian mixture distribution, it is known that the evolution of a finite family of summary statistics under stochastic gradient descent converges to an autonomous ordinary differential equation (ODE), as the dimension and sample size go to $\infty$ and the step size goes to $0$ commensurately. Our main result is that these ODE limits are universal in that this limit is the same whenever the data is drawn from mixtures of arbitrary product distributions whose first two moments match the corresponding Gaussian distribution, provided the initialization and ground truth vectors are coordinate-delocalized. We complement this by proving two corresponding non-universality results. We provide a simple example where the ODE limits are non-universal if the initialization is coordinate aligned. We also show that the stochastic differential equation limits arising as fluctuations of the summary statistics around their ODE's fixed points are not universal.

stat.ML↗

arXiv2025

Fixed-magnetization Ising on random graphs up to reconstruction

We study the fixed-magnetization ferromagnetic Ising model on random $d$-regular graphs for $d\ge 3$ and inverse temperature below the tree reconstruction threshold. Our main result is that for each magnetization $η$, the free energy density of the fixed-magnetization Ising model converges to the annealed free energy density, itself the Bethe free energy of an Ising measure on the infinite $d$-regular tree. Moreover, the fixed-magnetization Ising model exhibits local weak convergence to this tree measure. A key challenge to proving these results is that for magnetizations between the model's spinodal points, the limiting tree measure corresponds to an unstable fixed point of the belief propagation equations. As an application, we prove that the positive-temperature Zdeborová--Boettcher conjecture on max-cut and min-bisection holds up to the reconstruction threshold: on the random $d$-regular graph, the expected fraction of bichromatic edges in the anti-ferromagnetic Ising model plus the expected fraction of bichromatic edges in the zero-magnetization ferromagnetic Ising model equals $1+o(1)$. A second application is completely determining the large deviation rate function for the magnetization in the Ising model on the random regular graph up to reconstruction. Finally, we use the precise understanding of this rate function to show that the Glauber dynamics for the full Ising model on the random graph mixes in sub-exponential time from uniformly random initialization, well into the non-uniqueness regime where the worst-case initialization mixing time is exponentially slow.

math.PR↗

arXiv2025

Logarithmic delocalization of low temperature 3D Ising and Potts interfaces above a hard floor

We study the entropic repulsion of the low temperature 3D Ising and Potts interface in an $n\times n \times n$ box with blue boundary conditions on its bottom face (the hard floor), and red boundary conditions on its other five faces. For Ising, Frohlich and Pfister proved in 1987 that the typical interface height above the origin diverges (non-quantitatively), via correlation inequalities special to the Ising model; no such result was known for Potts. We show for both the Ising and Potts models that the entropic repulsion fully overcomes the potentially attractive interaction with the floor, and obtain a logarithmically diverging lower bound on the typical interface height. This is complemented by a conjecturally sharp upper bound of $\lfloor ξ^{-1}\log n\rfloor$ where $ξ$ is the rate function for a point-to-plane non-red connection under the infinite volume red measure. The proof goes through a coupled random-cluster interface to overcome the potential attractive interaction with the boundary, and a coupled fuzzy Potts model to reduce the upper bound to a simpler setting where the repulsion is attained by conditioning a no-floor interface to lie in the upper half-space.

math.PR↗

arXiv2025

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

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, e.g., 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 1-dimensional random processes, followed by analysis of the latter's escape from saddle points separating stable modes.

math.PR↗

arXiv2025

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

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. In this paper, we show that for a broad class of self-correcting quantum memories on lattices with parity check redundancies, a quasi-local quantum Gibbs sampler rapidly converges to the corresponding low-temperature Gibbs state within a logical sector when initialized from a ground state. This illustrates a dynamical view of self-correcting quantum memories, where the "syndrome sector" rapidly converges to thermal equilibrium, while the "logical sector" remains metastable. As a key application, when initialized from a random ground state, this gives a rapid Gibbs state preparation algorithm for the 4D toric code in $\mathrm{polylog}(n)$ depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.

quant-ph↗

arXiv2025

Rapid phase ordering for Ising and Potts dynamics on random regular graphs

We consider the Ising, and more generally, $q$-state Potts Glauber dynamics on random $d$-regular graphs on $n$ vertices at low temperatures $β\gtrsim \frac{\log d}{d}$. The mixing time is exponential in $n$ due to a bottleneck between $q$ dominant phases consisting of configurations in which the majority of vertices are in the same state. We prove that for any $d\ge 7$, from biased initializations with $ε_d n$ more vertices in state-$1$ than in other states, the Glauber dynamics quasi-equilibrates to the stationary distribution conditioned on having plurality in state-$1$ in optimal $O(\log n)$ time. Moreover, the requisite initial bias $ε_d$ can be taken to zero as $d \to \infty$. Even for the $q=2$ Ising case, where the states are naturally identified with $\pm 1$, proving such a result requires a new approach in order to control negative information spread in spacetime despite the model being in low temperature and exhibiting strong local correlations. For this purpose, we introduce a coupled non-Markovian rigid dynamics for which a delicate temporal recursion on probability mass functions of minus spacetime cluster sizes establishes their subcriticality.

math.PR↗

arXiv2025

Spectral alignment of stochastic gradient descent for high-dimensional classification tasks

We rigorously study the relation between the training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, both the SGD trajectory and emergent outlier eigenspaces of the Hessian and gradient matrices align with a common low-dimensional subspace. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

cs.LG↗

arXiv2025

Finding planted cliques using gradient descent

The planted clique problem is a paradigmatic model of statistical-to-computational gaps: the planted clique is information-theoretically detectable if its size $k\ge 2\log_2 n$ but polynomial-time algorithms only exist for the recovery task when $k= Ω(\sqrt{n})$. By now, there are many algorithms that succeed as soon as $k = Ω(\sqrt{n})$. Glaringly, however, no black-box optimization method, e.g., gradient descent or the Metropolis process, has been shown to work. In fact, Chen, Mossel, and Zadik recently showed that any Metropolis process whose state space is the set of cliques fails to find any sub-linear sized planted clique in polynomial time if initialized naturally from the empty set. We show that using the method of Lagrange multipliers, namely optimizing the Hamiltonian given by the sum of the objective function and the clique constraint over the space of all subgraphs, succeeds. In particular, we prove that Markov chains which minimize this Hamiltonian (gradient descent and a low-temperature relaxation of it) succeed at recovering planted cliques of size $k = Ω(\sqrt{n})$ if initialized from the full graph. Importantly, initialized from the empty set, the relaxation still does not help the gradient descent find sub-linear planted cliques. We also demonstrate robustness of these Markov chain approaches under a natural contamination model.

cs.DS↗

arXiv2025

Metastability in Glauber dynamics for heavy-tailed spin glasses

We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an $α$-stable law for $α\in (0,1)$. We show a sharp description of metastability on exponential timescales, in a form that is believed to hold for Glauber/Langevin dynamics for many mean-field spin glass models, but only known rigorously for the Random Energy Models. Namely, we establish a decomposition of the state space into sub-exponentially many wells, and show that the projection of the Glauber dynamics onto which well it resides in, asymptotically behaves like a Markov chain on wells with certain explicit transition rates. In particular, mixing inside wells occurs on much shorter timescales than transit times between wells, and the law of the next well the Glauber dynamics will fall into depends only on which well it currently resides in, not its full configuration. We can deduce consequences like an exact expression for the two-time autocorrelation functions that appear in the activated aging literature.

math.PR↗

arXiv2024

Low-temperature Ising dynamics with random initializations

It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be much faster. However, such phenomena have largely evaded rigorous analysis, as most tools in the study of Markov chain mixing times are tailored to worst-case initializations. In this paper we develop a general framework towards establishing this conjectured behavior for the Ising model. In the classical setting of the Ising model on an $N$-vertex torus in $\mathbb Z^d$, our framework implies that the mixing time for the Glauber dynamics, initialized from a $\frac 12$-$\frac 12$ mixture of the all-plus and all-minus configurations, is $N^{1+o(1)}$ in dimension $d=2$, and at most quasi-polynomial in all dimensions $d\ge 3$, at all temperatures below the critical one. The key innovation in our analysis is the introduction of the notion of "weak spatial mixing within a phase", a low-temperature adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to control the mixing time from the above random initialization (by relating it to the mixing time with plus boundary condition at $O(\log N)$ scales), and that it holds at all low temperatures in all dimensions. This framework naturally extends to much more general families of graphs. To illustrate this, we also use the same approach to establish optimal $O(N\log N)$ mixing for the Ising Glauber dynamics on random regular graphs at sufficiently low temperatures, when initialized from the same random mixture.

math.PR↗

arXiv2024

On the tractability of sampling from the Potts model at low temperatures via random-cluster dynamics

Sampling from the $q$-state ferromagnetic Potts model is a fundamental question in statistical physics, probability theory, and theoretical computer science. On general graphs, this problem may be computationally hard, and this hardness holds at arbitrarily low temperatures. At the same time, in recent years, there has been significant progress showing the existence of low-temperature sampling algorithms in various specific families of graphs. Our aim in this paper is to understand the minimal structural properties of general graphs that enable polynomial-time sampling from the $q$-state ferromagnetic Potts model at low temperatures. We study this problem from the perspective of random-cluster dynamics. These are non-local Markov chains that have long been believed to converge rapidly to equilibrium at low temperatures in many graphs. However, the hardness of the sampling problem likely indicates that this is not even the case for all bounded degree graphs. Our results demonstrate that a key graph property behind fast or slow convergence time for these dynamics is whether the independent edge-percolation on the graph admits a strongly supercritical phase. By this, we mean that at large $p<1$, it has a large linear-sized component, and the graph complement of that component is comprised of only small components. Specifically, we prove that such a condition implies fast mixing of the random-cluster Glauber and Swendsen--Wang dynamics on two general families of bounded-degree graphs: (a) graphs of at most stretched-exponential volume growth and (b) locally treelike graphs. In the other direction, we show that, even among graphs in those families, these Markov chains can converge exponentially slowly at arbitrarily low temperatures if the edge-percolation condition does not hold.

math.PR↗

arXiv2024

Fast relaxation of the random field Ising dynamics

We study the convergence properties of Glauber dynamics for the random field Ising model (RFIM) with ferromagnetic interactions on finite domains of $\mathbb{Z}^d$, $d \ge 2$. Of particular interest is the Griffiths phase where correlations decay exponentially fast in expectation over the quenched disorder, but there exist arbitrarily large islands of weak fields where low-temperature behavior is observed. Our results are twofold: 1. Under weak spatial mixing (boundary-to-bulk exponential decay of correlations) in expectation, we show that the dynamics satisfy a weak Poincaré inequality implying algebraic relaxation to equilibrium over timescales polynomial in the volume $N$ of the domain, and polynomial time mixing from a warm start. From this we construct a polynomial-time approximate sampling algorithm based on running Glauber dynamics over an increasing sequence of approximations of the domain. 2. Under strong spatial mixing (exponential decay of correlations even near boundary pinnings) in expectation, we prove a full Poincaré inequality, implying exponential relaxation to equilibrium and $N^{o(1)}$-mixing time. Note by way of example, both weak and strong spatial mixing hold at any temperature, provided the external fields are strong enough. Our proofs combine a stochastic localization technique which has the effect of increasing the variance of the field, with a field-dependent coarse graining which controls the resulting sub-critical percolation process of sites with weak fields.

math.PR↗