Yuval Peres

We analyze the asynchronous version of the DeGroot dynamics: In a connected graph G$$ G $$ with n$$ n $$ nodes, each node has an initial opinions in [0,1]$$ \left[0,1\right] $$ and an independent Poisson clock. When a clock at a node v$$ v $$ rings, the opinion at v$$ v $$ is replaced by the average opinion of its neighbors. It is well known that t...

We consider a boundary value problem for the p-Laplacian, posed in the exterior of small cavities that all have the same p-capacity and are anchored to the unit sphere in \(\mathbb {R}^d\), where \(1<p<d.\) We assume that the distance between anchoring points is at least \(\varepsilon \) and the characteristic diameter of cavities is \(\alpha \vare...

On a regular graph $G=(V,E)$, consider a function $\operatorname{val} \colon V \longrightarrow \{0,1\}$ that labels the vertices of $G$, and let $(X_i)$ be the simple random walk on $G$ started from a vertex chosen uniformly at random. In this work, we show that the Hamming weight of the sequence of bits $(\operatorname{val}(X_i))$ satisfies a loca...

We analyze the asynchronous version of the DeGroot dynamics: In a connected graph $G$ with $n$ nodes, each node has an initial opinion in $[0,1]$ and an independent Poisson clock. When a clock at a node $v$ rings, the opinion at $v$ is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We s...

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any C1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{am...

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs...

We consider a boundary value problem for the $p$-Laplacian, posed in the exterior of small cavities that all have the same $p$-capacity and are anchored to the unit sphere in $\mathbb{R}^d$, where $1<p<d.$ We assume that the distance between anchoring points is at least $\varepsilon$ and the characteristic diameter of cavities is $\alpha \varepsilo...

We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$ , where $a \geq 1$ is the system gain, $Z_n$ 's are independent random variables with bounded $\alpha$ -th moments, and $U_n$ 's are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as...

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random...

In the trace reconstruction problem, the goal is to reconstruct an unknown string $x$ of length $n$ from multiple traces obtained by passing $x$ through the deletion channel. In the relaxed problem of $approximate$ trace reconstruction, the goal is to reconstruct an approximation $\widehat{x}$ of $x$ which is close (within $\epsilon n$) to $x$ in e...

Let $P$ be a bistochastic matrix of size $n$, and let $\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\Pi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $\Pi$. We show that if th...

We study the stabilization of a linear control system with an unbounded random system gain where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system Xn+1 = AnXn+Wn-Un, where the An’s are drawn independently at random at each time n from a known distribution with unbounded support, and whe...

Benjamini et al. (Inst Hautes Études Sci Publ Math 90:5–43, 2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability 𝜖, the probability p𝜖 that the weighted majority changes is at most C𝜖1∕4. They asked what is the best possible exponent that could repla...

Population protocols are a fundamental model in distributed computing, where many nodes with bounded memory and computational power have random pairwise interactions over time. This model has been studied in a rich body of literature aiming to understand the tradeoffs between the memory and time needed to perform computational tasks. We study the p...

We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to infinity is the Poisson(1) branching process conditioned to survive forever. An extension to "almost" regular graphs and a quenched version are also given.

1963-2012 IEEE. Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p in P or from a distribution q in Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be ac...

We analyze a Markov chain, known as the product replacement chain, on the set of generating n-tuples of a fixed finite group G. We show that as \(n \rightarrow \infty \), the total-variation mixing time of the chain has a cutoff at time \(\frac{3}{2} n \log n\) with window of order n. This generalizes a result of Ben-Hamou and Peres (who establishe...

Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p ∈ P or from a distribution q ∈ Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple...

We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $\mu=\mu_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A....

Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p ∈ P or from a distribution q ∈ Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple...

Motivated by applications in wireless networks and the Internet of Things, we consider a model of n nodes trying to reach consensus with high probability on their majority bit. Each node i is assigned a bit at time 0 and is a finite automaton with m bits of memory (i.e., 2m states) and a Poisson clock. When the clock of i rings, i can choose to com...

Abstarct For a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n -vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log...

We provide an algorithm, running in polynomial time in the number of vertices, computing the unique solution to the biased infinity Laplacian Boundary Problem on finite graphs. The algorithm is based on the general outline and approach taken in the corresponding algorithm for the unbiased case provided by Lazarus et al. The new ingredient is an adj...

A special case of Myerson's classic result describes the revenue-optimal equilibrium when a seller offers a single item to a buyer. We study a repeated sales extension of this model: a seller offers to sell a single fresh copy of an item to the same buyer every day via a posted price. The buyer's private value for the item is drawn initially from a...

Let $T_\lambda$ be a Galton--Watson tree with Poisson($\lambda$) offspring, and let $A$ be a tree property. In this paper, are concerned with the regularity of the function $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $A_k$, depending only on the f...

The stochastic multi-armed bandit problem is a classic model illustrating the tradeoff between exploration and exploitation. We study the effects of competition and cooperation on this tradeoff. Suppose there are $k$ arms and two players, Alice and Bob. In every round, each player pulls an arm, receives the resulting reward, and observes the choice...

A Web crawler is an essential part of a search engine that procures information subsequently served by the search engine to its users. As the Web is becoming increasingly more dynamic, in addition to discovering new web pages a crawler needs to keep revisiting those already in the search engine's index, in order to keep the index fresh by picking u...

We consider the sorted top-$k$ problem whose goal is to recover the top-$k$ items with the correct order out of $n$ items using pairwise comparisons. In many applications, multiple rounds of interaction can be costly. We restrict our attention to algorithms with a constant number of rounds $r$ and try to minimize the sample complexity, i.e. the num...

We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain Ω⊂RN. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: “Those domains whose uni...

An algorithm observes the trajectories of random walks over an unknown graph G , starting from the same vertex x , as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges m up to a bounded factor in O(t_{\mathrm {rel}}^{3/4}\sqrt{m/d}\,) steps, where t_{\mathrm {rel}} is the relaxation ti...

We consider the non-stochastic version of the (cooperative) multi-player multi-armed bandit problem. The model assumes no communication at all between the players, and furthermore when two (or more) players select the same action this results in a maximal loss. We prove the first $\sqrt{T}$-type regret guarantee for this problem, under the feedback...

Significance Online learning is a prominent learning paradigm where data become available over time. The main challenge is to predict future events using past observations and any domain-specific knowledge available, such as expert advice. We study multiclass online learning, where a forecaster is choosing each day one of d options based on the adv...

Suppose that ${\mathcal G}$ is a finite, connected graph and $X$ is a lazy random walk on ${\mathcal G}$. The lamplighter chain $X^\diamond$ associated with $X$ is the lazy random walk on the wreath product ${\mathcal G}^\diamond = {\mathbb Z}_2 \wr {\mathcal G}$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $\underline{f}$ i...

Motivated by applications in blockchains and sensor networks, we consider a model of $n$ nodes trying to reach consensus on their majority bit. Each node $i$ is assigned a bit at time zero, and is a finite automaton with $m$ bits of memory (i.e., $2^m$ states) and a Poisson clock. When the clock of $i$ rings, $i$ can choose to communicate, and is t...

Given a subgraph $H$ of a graph $G$, the induced graph of $H$ is the largest subgraph of $G$ whose vertex set is the same as that of $H$. Our paper concerns the induced graphs of the components of $\operatorname{WSF}(G)$, the wired spanning forest on $G$, and, to a lesser extent, $\operatorname{FSF}(G)$, the free uniform spanning forest. We show th...

We consider an unstable scalar linear stochastic system, X_(n + 1) = aX_n + Z_n – U_n.; where a ≥ 1 is the system gain, Z_n's are independent random variables with bounded α-th moments, and U_n'S are the control actions that are chosen by a controller who receives a single element of a finite set {1, …, M} as its only information about system state...

We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain $\Omega\subset\mathbb{R}^N$. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: "...

Significance Given a set L of n points on the sphere, an allocation is a way to divide the sphere into n cells of equal area, each associated with a point of L . Given two sets of n points A and B on the sphere, a matching is a bijective map from A to B . Allocation and matching rules that minimize the distance between matched points are related to...

We study a graph-theoretic model of interface dynamics called $Competitive\, Erosion$. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy...

We consider the problem of testing graph cluster structure: given access to a graph $G=(V, E)$, can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish...

We study the multiclass online learning problem where a forecaster makes a sequence of predictions using the advice of $n$ experts. Our main contribution is to analyze the regime where the best expert makes at most $b$ mistakes and to show that when $b = o(\log_4{n})$, the expected number of mistakes made by the optimal forecaster is at most $\log_...

Significance We present a tractable algorithm that provides a near-optimal solution to the crawling problem, a fundamental challenge at the heart of web search: Given a large quantity of distributed and dynamic web content, what pages do we choose to update a local cache with the goal of serving up-to-date pages to client requests? Solving this opt...

We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$, where $a \geq 1$ is the system gain, $Z_n$'s are independent random variables with bounded $\alpha$-th moments, and $U_n$'s are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as its only informa...

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a discrete-time version of the first-named author's ``Meeting time lemma"~ that bounds the probability of random wal...

We consider simple random walks on two partially directed square lattices. One common feature of these walks is that they are bound to revolve clockwise; however they exhibit different recurrence/transience behaviors. Our main result is indeed a proof of recurrence for one of the graphs, solving a conjecture of Menshikov et al. ('17). For the other...

2017, University of Washington. All rights reserved. We consider two independent Markov chains on the same finite state space, and study their intersection time, which is the first time that the trajectories of the two chains intersect. We denote by tI the expectation of the intersection time, maximized over the starting states of the two chains. W...

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X_(n+1) = A_n X_n +W_n –U_n, where the A_n's are drawn independently at random at each time n from a known distribution with unbounded support, and where the cont...

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X_(n+1) = A_nX_n + W_n – U_n, where the A_n's are drawn independently at random at each time n from a known distribution with unbounded support, and where the con...

We analyze a Markov chain, known as the product replacement chain, on the set of generating $n$-tuples of a fixed finite group $G$. We show that as $n \rightarrow \infty$, the total-variation mixing time of the chain has a cutoff at time $\frac{3}{2} n \log n$ with window of order $n$. This generalizes a result of Ben-Hamou and Peres (who establish...

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system $X_{n+1} = A_n X_n + W_n - U_n$, where the $A_n$'s are drawn independently at random at each time $n$ from a known distribution with unbounded support, and where...

We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.

The insertion-deletion channel takes as input a bit string \mathbf x \in \{0,1\}^{n} , and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover \mathbf x from many independent outputs (called "traces") of the insertion-deletion channel applied to \mathbf x . We show that...

A sequence of chains exhibits (total-variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon< 1/2$, the ratio $t_{\mathrm{mix}}^{(n)}(\epsilon)/t_{\mathrm{mix}}^{(n)}(1-\epsilon)$ tends to 1 as $n \to \infty $ (resp., the $\limsup$ of this ratio is bounded uniformly in $\epsilon$), where $t_{\mathrm{mix}}^{(n)}(\epsilon)$ is the $\epsilon$-tot...

It is well known that sequential decision making may lead to information cascades. That is, when agents make decisions based on their private information, as well as observing the actions of those before them, then it might be rational to ignore their private signal and imitate the action of previous individuals. If the individuals are choosing bet...

It is well known that sequential decision making may lead to information cascades. That is, when agents make decisions based on their private information, as well as observing the actions of those before them, then it might be rational to ignore their private signal and imitate the action of previous individuals. If the individuals are choosing bet...

Let $(X_t)_{t = 0 }^{\infty}$ be an irreducible reversible discrete-time Markov chain on a finite state space $\Omega $. Denote its transition matrix by $P$. To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain $(X_t^{\mathrm{c}})_{t \ge 0} $ whose kernel is given by...

In this paper we discuss the problem of estimating graph parameters from a random walk with restarts at a fixed vertex $x$. For regular graphs $G$, one can estimate the number of vertices $n_G$ and the $\ell^2$ mixing time of $G$ from $x$ in $\widetilde{O}(\sqrt{n_G}\,(t_{\rm unif}^G)^{3/4})$ steps, where $t_{\rm unif}^G$ is the uniform mixing time...

The spectral gap $\gamma$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $\gamma$ from this data. Let $\p...

The spectral gap $\gamma$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $\gamma$ from this data. Let $\p...

We consider the stable matching of two independent Poisson processes in $\mathbb{R}^d$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We...

In the trace reconstruction problem an unknown string ${\bf x}=(x_0,\dots,x_{n-1})\in\{0,1,...,m-1\}^n$ is observed through the deletion channel, which deletes each $x_k$ with a certain probability, yielding a contracted string $\widetilde{\bf X}$. Earlier works have proved that if each $x_k$ is deleted with the same probability $q\in[0,1)$, then $...

Hoffman [7] proved a matrix inequality that yields a useful upper bound on the number of walks in a graph. Sidorenko [14] extended the bound on the number of walks to a bound on the number of homomorphisms from a tree to a graph. In this expository note, we give a short probabilistic proof of both results, using the basic identity of importance sam...

The deletion channel takes as input a bit string $\mathbf{x} \in \{0,1\}^n$, and deletes each bit independently with probability $q$, yielding a shorter string. The trace reconstruction problem is to recover an unknown string $\mathbf{x}$ from many independent outputs (called "traces") of the deletion channel applied to $\mathbf{x}$. We show that i...

We consider random walk on dynamical percolation on the discrete torus $\mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(\mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynam...

We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $\mu=\mu_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A....

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixi...

In 1988, Johnson, Papadimitriou and Yannakakis wrote that “Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems". Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and ind...

In the trace reconstruction problem, an unknown bit string x ∈ {0,1}ⁿ is observed through the deletion channel, which deletes each bit of x with some constant probability q, yielding a contracted string x. How many independent copies of x are needed to reconstruct x with high probability? Prior to this work, the best upper bound, due to Holenstei...

In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they...

Let \({(G,\rho)}\) be a stationary random graph, and use \({B^G_{\rho}(r)}\) to denote the ball of radius r about \({\rho}\) in G. Suppose that \({(G,\rho)}\) has annealed polynomial growth, in the sense that \({\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)}\) for some \({k > 0}\) and every \({r \geq 1}\). Then there is an infinite sequence of times \({\...

We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) as a special case of concentration for convex Lipschitz functions. For $d=2$, exponential concentration was shown by Marton (20...

We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related setti...

We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3}{2}n\log n$ with window of size $n$, solving a question posed by Chung and Graha...

We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3}{2}n\log n$ with window of size $n$, solving a question posed by Chung and Graha...

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of st...

We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius $r$ is $O(r^3)$. We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any...

We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius $r$ is $O(r^{5/2})$.. We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside...

In this note we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Also a few Beurling type hitting estimates are obtained for the random walk on discretizations of smooth domains. These have been used recently in the study of a two dimensional competing aggregation system kno...

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains.We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of sta...