arXiv subjects

Allan Sly

Publications and source records attributed to Allan Sly.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the number and size of Markov equivalence classes of random directed acyclic graphs

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 decomposition, we show that the ratio between the number of Markov equivalence classes and directed acyclic graphs approaches a positive constant when the number of sites goes to infinity. For a typical directed acyclic graph, the expected number of elements in its Markov equivalence class remains bounded. More precisely, we prove that for a uniformly chosen directed acyclic graph, the size of its Markov equivalence class has super-polynomial tails.

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arXiv2024

Infinite cycles in the interchange process in five dimensions

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 interchange. In this way, a random permutation $π_β:V\to V$ is formed for any time $β>0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $β$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.

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arXiv2023

Rotationally invariant first passage percolation: Concentration and scaling relations

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 under hypotheses which can be verified for many standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. This is the first such tight concentration result known for any model that is not exactly solvable. As a consequence, we prove a version of the so called KPZ relation between the passage time fluctuations and the transversal fluctuations of geodesics as well as up to constant upper and lower bounds for the non-random fluctuations in these models. Similar results have previously been known conditionally under unproven hypotheses, but our results are the first ones that apply to some specific FPP models. Our arguments are expected to be useful in proving a number of other estimates which were hitherto only known conditionally or for exactly solvable models.

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arXiv2023

Optimal reconstruction of general sparse stochastic block models

This paper is motivated by the reconstruction problem on the sparse stochastic block model. Mossel, et. al. proved that a reconstruction algorithm that recovers an optimal fraction of the communities in the symmetric, 2-community case. The main contribution of their proof is to show that when the signal to noise ratio is sufficiently large, in particular $λ^2d > C$, the reconstruction accuracy for a broadcast process on a tree with or without noise on the leaves is asymptotically the same. This paper will generalize their results, including the main step, to a general class of the sparse stochastic block model with any number of communities that are not necessarily symmetric, proving that an algorithm closely related to Belief Propagation recovers an optimal fraction of community labels.

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arXiv2023

One-step replica symmetry breaking of random regular NAE-SAT II

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 the result to probability arbitrarily close to one. To do so, we introduce a new framework which is the synthesis of two approaches: the small subgraph conditioning and a variance decomposition technique using Doob martingales and discrete Fourier analysis. The main challenge is a delicate integration of the two methods to overcome the difficulty arising from applying the moment method to an unbounded state space.

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arXiv2023

On a random model of forgetting

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 $S$ one by one, in order. When an entering element is larger than the current minimum element of $S$, this minimum leaves $S$. Let $S(1,n)$ denote the content of $S$ after the first $n$ elements $x_k$ join. Simulations suggest that the size $|S(1,n)|$ of $S$ at time $n$ is typically close to $n/e$. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ is of size at most $\tilde{O}(\sqrt n)$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as $n$ tends to infinity $ n^{-1/2}\big( |S(1,n)|-n/e \big) $ converges to a normal random variable with variance $3e^{-2}-e^{-1}$. We further show that the dynamics of the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ converges with proper scaling to a three dimensional Bessel process.

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arXiv2023

One-step replica symmetry breaking of random regular NAE-SAT I

In a broad class of sparse random constraint satisfaction problems(CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPs, it was verified \cite{ssz22} that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condenses on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability bounded away from zero, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We believe that our method is applicable to a broad range of random CSPs in the 1RSB universality class.

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arXiv2023

The number of solutions for random regular NAE-SAT

Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain "survey propagation model" associated to this problem. We believe that these methods may be applicable in a wide class of related problems.

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arXiv2023

Infinite order phase transition in the slow bond TASEP

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 whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of $\varepsilon$ is 0. This was ultimately resolved rigorously in Basu-Sidoravicius-Sly which established that $\varepsilon_c=0$. Here we study the effect of the current as $\varepsilon$ tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when $\varepsilon$ is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal.

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arXiv2022

The SIR model in a moving population: propagation of infection and herd immunity

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 are removed at rate $ν$. We show that when $ν$ is small, with positive probability the infection survives forever and grows linearly. Furthermore, after the infection reaches a region, it quickly passes through and leaves behind a $\textit{herd immunity}$ regime consisting of recovered particles, a small positive density of susceptible particles, and no infected particles. One notable feature of this model is the simultaneously existence of supercritical and subcritical phases on either side of an infection front of $O(1)$ width.

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arXiv2022

Spread of infections in a heterogeneous moving population

We consider a model where an infection moves through a collection of particles performing independent random walks. In this model, Kesten and Sidoravicius established linear growth of the infected region when infected and susceptible particles move at the same speed. In this paper we establish a linear growth rate when infected and susceptible particles move at different speeds, answering an open problem from their work. Our proof combines an intricate coupling of Poisson processes with a streamlined version of a percolation model of Sidoravicius and Stauffer.

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arXiv2022

Ising model on trees and factors of IID

We study the ferromagnetic Ising model on the infinite $d$-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature $β\ge 0$ satisfies $\tanh β\le (d-1)^{-1}$. However, in the reconstruction regime ($\tanh β> (d-1)^{-\frac{1}{2}}$), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons. The solution $\{X_t(v) \}$ of the SDE is distributed as \[ X_t(v) = tτ_v + B_t(v), \] where $\{τ_v \}$ is an Ising sample and $\{B_t(v) \}$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $\tanh β\le c(d-1)^{-\frac{1}{2}}$, where $c>0$ is an absolute constant.

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arXiv2022

Stationary Distributions for the Voter Model in $d\geq 3$ are Factors of IID

For the Voter Model on $\mathbb{Z}^d$, $d\geq 3$, we show that the (extremal) stationary distributions are isomorphic to Bernoulli shifts, and answer an open question asked by Steif and Tykesson. The proof gives explicit constructions of the stationary distributions as factors of IID processes on $\mathbb{Z}^d$.

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arXiv2021

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

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 to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognormal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

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arXiv2021

Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster

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 to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu '21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.

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arXiv2021

Universality of cutoff for graphs with an added random matching

We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph $G=(V,E)$ with $|V|$ even we define a random graph $ G^*=(V,E \cup E')$ obtained by picking $E'$ to be the (unordered) pairs of a random perfect matching of $V$. We show that for a sequence of such graphs $G_n$ of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of $G_n$ is at least 3 for all $n$, then the random walk on $G_n^*$ exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff.

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arXiv2021

Proof of the satisfiability conjecture for large k

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $α_*(k)$ such that a random $k$-SAT formula of clause density $α$ is with high probability satisfiable for $α<α_*$, and unsatisfiable for $α>α_*$. We show that the threshold $α_*(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class.

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arXiv2021

A phase transition for repeated averages

Let $x_1,\ldots,x_n$ be a fixed sequence of real numbers. At each stage, pick two indices $I$ and $J$ uniformly at random and replace $x_I$, $x_J$ by $(x_I+x_J)/2$, $(x_I+x_J)/2$. Clearly all the coordinates converge to $(x_1+\cdots+x_n)/n$. We determine the rate of convergence, establishing a sharp "cutoff" transition, answering a question of Jean Bourgain.

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