arXiv subjects

Allan Sly

Publications and source records attributed to Allan Sly.

At least 19 recordsLinked to original sources

arXiv2026

Cutoff for Simple Random Walk on Preferential Attachment Graphs

We consider simple random walk on the sequential affine preferential attachment graphs with fixed parameters $m\ge2$ and $δ\ge0$. We prove the walk has total-variation cutoff, and for every fixed $\varepsilon\in(0,1)$, \[ t_{\mathrm{mix}}^{(n)}(\varepsilon) =\frac{\log n}{νd} +O_{\mathbb P}\!\left(\sqrt{\log n}\right), \] where $ν>0$ is the walk speed and $d>0$ is the entropy per level of harmonic measure on the Pólya--point tree, the local weak limit of the graph. Thus cutoff occurs at the entropic time. Entropic cutoff was previously established for a range of random graph models including configuration models under mild degree assumptions and on Erdős--Rényi random graphs for typical starting points. We extend the harmonic-measure approach to preferential attachment, accounting for its age-dependent structure and hubs of polynomial degree. The local weak limit, the Pólya--point tree, provides a more tractable setting for analyzing the walk. The proof combines regeneration estimates on the local limit tree, a quantitative graph--tree coupling on the mixing-time scale, and control of hub excursions. We show that shortly before the cutoff time its distribution is close in total variation distance to a measure with small $L^\infty$ distance from stationarity. When $m\ge2$, the spectral gap is bounded away from zero with high probability which completes the upper-bound coupling argument.

math.PR↗

arXiv2026

Information Percolation and cutoff for Potts Glauber Dynamics

We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $Λ_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. More precisely, for every fixed $d\ge2$ and $q\ge3$, there exists $β_0=β_0(d,q)>0$ such that, for $0<β<β_0$, the cutoff occurs in an optimal $O(1)$ window centered at \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |Λ_n|, \] where $\mathfrak{r}\in(0,1)$ is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive $O(1)$. It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.

math.PR↗

arXiv2026

Improved rapid phase ordering for Ising dynamics on random regular graphs

We study Glauber dynamics for the Ising model on random $d$-regular graphs with $n$ vertices at low temperature, in the regime $β\geq C/d$ where $C,d$ are sufficiently large constants. In this regime, the full mixing time is exponential in $n$, because of the bottleneck between the two dominant phases: configurations with a majority of plus spins and configurations with a majority of minus spins. We show that, for sufficiently large $d$, if the dynamics is initialized with an excess of at least $ε_{C,d} n$ plus spins over minus spins, then it quasi-equilibrates in time $O(\log n)$ to the Gibbs measure conditioned on positive magnetization. In addition, the required bias $ε_{C,d}$ can be chosen so that $ε_{C,d}\to 0$ as $d,C\to\infty$. Previous bound of \cite{GSS25+} in the Ising and large-$d$ setting required $β\gtrsim (\log d)/d$. A key difficulty is that, even from the all-plus initialization, one must control the propagation of negative information in spacetime in a low-temperature model with strong local correlations. We address this by developing a new bootstrap approach to proving subcriticality of disagreement clusters under the grand coupling.

math.PR↗

arXiv2026

Random walk on the small-world network model in 3 or more dimensions

We study the mixing time of a simple random walk on the small-world network defined by adding edges to $\mathbb{Z}_n^d$ as follows: for each pair $\{x,y\}$ we add an edge with probability $Z_n/\|x-y\|^{d}$ with $Z_n$ chosen so that the average number of added edges to every vertex is $1$. When $d\geq 3$, we show that with high probability the mixing time is of order $\log n$ and that the random walk does not exhibit cutoff.

math.PR↗

arXiv2026

A Hierarchical Language Model with Predictable Scaling Laws and Provable Benefits of Reasoning

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with \emph{any} valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive \emph{reasoning} model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.

cs.LG↗

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

Rotationally invariant first passage percolation: Breaking the $n/\log n$ variance barrier

For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly smaller than $1$. Following Kesten's $O(n)$ upper bound (Ann. Appl. Probab., 1993) on the variance, Benjamini, Kalai and Schramm (Ann. Probab., 2003) used hypercontractivity to obtain an improvement of a factor of $\log n$ when passage times take two values with equal probability. This was later extended to more general classes of passage time distributions. However, unlike in exactly solvable planar models in last passage percolation where the variance is known to be $Θ(n^{2/3})$, the best known upper bound for the variance of passage times has remained $O(n/\log n)$ in all non-trivial variants of FPP. For a class of rotationally invariant Riemannian FPP on the plane, we show that the variance is $O(n^{1-\varepsilon})$ for some $\varepsilon>0$. Our argument uses fluctuation estimates for passage times and geodesics derived in Basu, Sidoravicius and Sly (2023) together with a multi-scale argument to establish that the geodesic exhibits disorder chaos, i.e., upon resampling a small fraction of the underlying randomness, the updated geodesic has on average a small overlap with the original one; this, established at a large number of scales, leads to a polynomial improvement of the variance bound.

math.PR↗

arXiv2026

Distribution of the magnetization of the critical Ising model on sparse random graphs

In this paper, we consider the Ising model on random $d$-regular graphs (with $d\ge3$) and Erdös-Rényi graphs $G(n,d/n)$ (with $d>1$) at the critical temperature. We prove that the \textit{magnetization}, i.e.\ the sum of the spins of a configuration, is typically of order $n^{3/4}$ and when multiplied by $n^{-3/4}$ converges in distribution to a non-trivial random variable, whose density we describe. In the regular graph case, the Small Subgraph Conditioning Method applies, and the limiting density is of the form $\frac1{Z}\exp(-C_d z^4)$. Surprisingly, in the Erdös-Rényi case, while the ratio of the second moment and first moment squared is bounded, the short cycle count is not enough to explain the fluctuations of the partition function restricted to a particular magnetization. We identify the additional source of randomness as path counts of slowly diverging length. This quantity is motivated by the heuristic that correlations between distant vertices are proportional to their local branching rate. Augmenting the Small Subgraph Conditioning Method with these path counts allows us to prove convergence of the magnetization to a non-deterministic limiting distribution. To our knowledge, the need to condition on graph observables beyond the cycle counts is a new phenomenon for spin systems. As further corollaries, we derive a polynomial lower bound on the mixing time of the stochastic Ising model on sparse random graphs at the critical temperature complementing recent upper bounds. Moreover, we establish the fluctuations of the free energy in the Erdös-Rényi case, answering a recent question of Coja-Oghlan et. al.

math.PR↗

arXiv2026

Mixing times for the TASEP on the circle

We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a circle of length $N$ with $k$ particles. We show that the mixing time is of order $N^2 \min(k,N-k)^{-1/2}$, and that the cutoff phenomenon does not occur. This confirms behavior which was separately predicted by Jara, Lacoin and Peres, and it is more broadly believed to hold for integrable models in the KPZ-universalty class. Our arguments rely on a connection to periodic last passage percolation with a detailed analysis of flat geodesics, as well as a novel random extension and time shift argument for last passage percolation.

math.PR↗

arXiv2025

Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs

We consider the stochastic Ising model on sparse Erdos-Renyi graphs $G(n,d/n)$ with $d>1$ at the critical temperature $β_c=\tanh^{-1}(d^{-1})$ and prove that with high probability, the mixing time is at most polynomial in $n$. Our approach combines the recent stochastic localization framework of Chen and Eldan, which yields spectral gap bounds in the well-behaved bulk of the graph, together with classical results on the relaxation time of Glauber dynamics on trees to handle regions where we cannot apply the Chen-Eldan method directly because of atypically large local neighborhoods.

math.PR↗

arXiv2025

Potts and random cluster measures on locally regular-tree-like graphs

Fixing $β\ge 0$ and an integer $q \ge 2$, consider the ferromagnetic $q$-Potts measures $μ_n^{β,B}$ on finite graphs ${\sf G}_n$ on $n$ vertices, with external field strength $B \ge 0$ and the corresponding random cluster measures $φ^{q,β,B}_{n}$. Suppose that as $n \to \infty$ the uniformly sparse graphs ${\sf G}_n$ converge locally to an infinite $d$-regular tree ${\sf T}_{d}$, $d \ge 3$. We show that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which has been proved in case $d$ is even, or when $B=0$), yields the local weak convergence of $φ^{q,β,B}_n$ and $μ_n^{β,B}$ to the corresponding free or wired random cluster measure, Potts measure, respectively, on ${\sf T}_{d}$. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing {as limit points on} the critical line $β_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $β>β_c$, we further establish a pure-state decomposition by showing that conditionally on the same dominant color $1 \le k \le q$, the $q$-Potts measures on such edge-expander graphs ${\sf G}_n$ converge locally to the $q$-Potts measure on ${\sf T}_{d}$ with a boundary wired at color $k$.

math.PR↗

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

Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborová (2011) regarding the block model, in particular in the case of $q=3$ and $q=4$ communities. We prove that for $q=3$ and $q=4$ there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. The proof is based on showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for $q=3$ and $q=4$ if the average degree is sufficiently large. This improves on the result of Sly (2009), who proved similar results for regular trees for $q=3$. Our analysis of the critical case $q=4$ provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic case depends on the average degree of the tree. We also prove that for $q\geq 5$, the Kestin-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborová (2011), Moore (2017), Abbe and Sandon (2018) and Ricci-Tersenghi, Semerjian, and Zdeborová (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.

math.PR↗

arXiv2025

Local geometry of NAE-SAT solutions in the condensation regime

The local behavior of typical solutions of random constraint satisfaction problems (CSP) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. The work of Coja-Oghlan, Kapetanopoulos, Müller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular NAE-SAT model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random CSPs in the one-step replica symmetry breaking (1RSB) class. Our result is non-asymptotic, and characterizes the tight fluctuation $O(n^{-1/2})$ around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1RSB belief-propagation fixed point. We believe that our proof technique has broad applicability to random CSPs in the 1RSB class.

math.PR↗

arXiv2025

Weak recovery, hypothesis testing, and mutual information in stochastic block models and planted factor graphs

The stochastic block model is a canonical model of communities in random graphs. It was introduced in the social sciences and statistics as a model of communities, and in theoretical computer science as an average case model for graph partitioning problems under the name of the ``planted partition model.'' Given a sparse stochastic block model, the two standard inference tasks are: (i) Weak recovery: can we estimate the communities with non trivial overlap with the true communities? (ii) Detection/Hypothesis testing: can we distinguish if the sample was drawn from the block model or from a random graph with no community structure with probability tending to $1$ as the graph size tends to infinity? In this work, we show that for sparse stochastic block models, the two inference tasks are equivalent except at a critical point. That is, weak recovery is information theoretically possible if and only if detection is possible. We thus find a strong connection between these two notions of inference for the model. We further prove that when detection is impossible, an explicit hypothesis test based on low degree polynomials in the adjacency matrix of the observed graph achieves the optimal statistical power. This low degree test is efficient as opposed to the likelihood ratio test, which is not known to be efficient. Moreover, we prove that the asymptotic mutual information between the observed network and the community structure exhibits a phase transition at the weak recovery threshold. Our results are proven in much broader settings including the hypergraph stochastic block models and general planted factor graphs. In these settings we prove that the impossibility of weak recovery implies contiguity and provide a condition which guarantees the equivalence of weak recovery and detection.

math.PR↗

arXiv2025

The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$

We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is $O(m^2n \log n+ n^2 m^{o(1)})$. This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley.

math.PR↗

arXiv2025

Likelihood-Based Root State Reconstruction on a Tree: Sensitivity to Parameters and Applications

We consider a broadcasting problem on a tree where a binary digit (e.g., a spin or a nucleotide's purine/pyrimidine type) is propagated from the root to the leaves through symmetric noisy channels on the edges that randomly flip the state with edge-dependent probabilities. The goal of the reconstruction problem is to infer the root state given the observations at the leaves only. Specifically, we study the sensitivity of maximum likelihood estimation (MLE) to uncertainty in the edge parameters under this model, which is also known as the Cavender-Farris-Neyman (CFN) model. Our main result shows that when the true flip probabilities are sufficiently small, the posterior root mean (or magnetization of the root) under estimated parameters (within a constant factor) agrees with the root spin with high probability and deviates significantly from it with negligible probability. This provides theoretical justification for the practical use of MLE in ancestral sequence reconstruction in phylogenetics, where branch lengths (i.e., the edge parameters) must be estimated. As a separate application, we derive an approximation for the gradient of the population log-likelihood of the leaf states under the CFN model, with implications for branch length estimation via coordinate maximization.

math.PR↗

arXiv2024

Subcritical epidemics on random graphs

We study the contact process on random graphs with low infection rate $λ$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $λ_c$. By contrast, on the Erdős-Rényi random graphs $\mathcal G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{λ^{2+o(1)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a shaper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.

math.PR↗