Allan Sly
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View article: Journal of the European Mathematical Society
Journal of the European Mathematical Society Open
We considerably improve upon the recent result of Martinelli and Toninelli on\nthe mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$\nat low temperature and with random boundary conditions whose distribution $P$\n…
View article: Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs
Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs Open
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…
View article: Rapid phase ordering for Ising and Potts dynamics on random regular graphs
Rapid phase ordering for Ising and Potts dynamics on random regular graphs Open
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 $q$…
View article: Convergence of the environment seen from geodesics in exponential last-passage percolation
Convergence of the environment seen from geodesics in exponential last-passage percolation Open
A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation o…
View article: Likelihood-Based Root State Reconstruction on a Tree: Sensitivity to Parameters and Applications
Likelihood-Based Root State Reconstruction on a Tree: Sensitivity to Parameters and Applications Open
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 …
View article: Polynomial Mixing of the critical Glauber Dynamics for the Ising Model
Polynomial Mixing of the critical Glauber Dynamics for the Ising Model Open
In this note, we prove that on any graph of maximal degree $d$ the mixing time of the Glauber Dynamics for the Ising Model at $β_c=\tanh^{-1}(\frac1{d-1})$, the uniqueness threshold on the infinite $d$-regular tree, is at most polynomial i…
View article: Local geometry of NAE-SAT solutions in the condensation regime
Local geometry of NAE-SAT solutions in the condensation regime Open
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 …
View article: Weak recovery, hypothesis testing, and mutual information in stochastic block models and planted factor graphs
Weak recovery, hypothesis testing, and mutual information in stochastic block models and planted factor graphs Open
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 part…
View article: Local Geometry of NAE-SAT Solutions in the Condensation Regime
Local Geometry of NAE-SAT Solutions in the Condensation Regime Open
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 co…
View article: One-Step Replica Symmetry Breaking of Random Regular NAE-SAT II
One-Step Replica Symmetry Breaking of Random Regular NAE-SAT II Open
Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I, arXiv:2011.14270 , 2020), we study the random regular k - nae-sat model in the condensation regime. In Nam et al. (2020), the (1 rsb…
View article: Potts and random cluster measures on locally regular-tree-like graphs
Potts and random cluster measures on locally regular-tree-like graphs Open
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,…
View article: Rotationally invariant first passage percolation: Concentration and scaling relations
Rotationally invariant first passage percolation: Concentration and scaling relations Open
For rotationally invariant first passage percolation (FPP) on the plane, we use a multi-scale argument to prove stretched exponential concentration of the first passage times at the scale of the standard deviation. Our results are proved u…
View article: Infinite order phase transition in the slow bond TASEP
Infinite order phase transition in the slow bond TASEP Open
In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz posed the well‐known question of whether such very small perturbation…
View article: Local geometry of NAE-SAT solutions in the condensation regime
Local geometry of NAE-SAT solutions in the condensation regime Open
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 co…
View article: Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees
Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees Open
In this paper, we 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.
View article: Learning sparse graphons and the generalized Kesten–Stigum threshold
Learning sparse graphons and the generalized Kesten–Stigum threshold Open
The problem of learning graphons has attracted considerable attention across several scientific communities, with significant progress over the re-cent years in sparser regimes. Yet, the current techniques still require diverg-ing degrees …
View article: Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees
Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees Open
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 $q=4$ communit…
View article: Infinite cycles in the interchange process in five dimensions
Infinite cycles in the interchange process in five dimensions Open
In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interc…
View article: The SIR model in a moving population: propagation of infection and herd immunity
The SIR model in a moving population: propagation of infection and herd immunity Open
In a collection of particles performing independent random walks on $\mathbb Z^d$ we study the spread of an infection with SIR dynamics. Susceptible particles become infected when they meet an infected particle. Infected particles heal and…
View article: On the number and size of Markov equivalence classes of random directed acyclic graphs
On the number and size of Markov equivalence classes of random directed acyclic graphs Open
In causal inference on directed acyclic graphs, the orientation of edges is in general only recovered up to Markov equivalence classes. We study Markov equivalence classes of uniformly random directed acyclic graphs. Using a tower decompos…
View article: Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster
Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster Open
It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically …
View article: Subcritical epidemics on random graphs
Subcritical epidemics on random graphs Open
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…
View article: Mixing times for the TASEP on the circle
Mixing times for the TASEP on the circle Open
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 occu…
View article: On a random model of forgetting
On a random model of forgetting Open
Georgiou, Katkov and Tsodyks considered the following random process. Let $x_1,x_2,\ldots $ be an infinite sequence of independent, identically distributed, uniform random points in $[0,1]$. Starting with $S=\{0\}$, the elements $x_k$ join…
View article: Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron
Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron Open
YY We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made…
View article: One-step replica symmetry breaking of random regular NAE-SAT II
One-step replica symmetry breaking of random regular NAE-SAT II Open
Continuing our earlier work in \cite{nss20a}, we study the random regular k-NAE-SAT model in the condensation regime. In \cite{nss20a}, the 1RSB properties of the model were established with positive probability. In this paper, we improve …
View article: Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster
Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster Open
It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically …
View article: Infinite order phase transition in the slow bond TASEP
Infinite order phase transition in the slow bond TASEP Open
In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to $1-\varepsilon$ for some small $\varepsilon>0$. Janowsky and Lebowitz posed the well-known question of wheth…
View article: Upper Tail Large Deviations in First Passage Percolation
Upper Tail Large Deviations in First Passage Percolation Open
For first passage percolation on with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at distance n is macroscopically larger than typi…