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Yuval Peres

Publications and source records attributed to Yuval Peres.

At least 127 records · Page 7Linked to original sources
arXiv2015

The divisible sandpile at critical density

The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses at most 1. The process stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1, where $m$ is the mean mass per vertex. The main result of this paper is that in the critical case m=1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field.

math.PR↗
arXiv2015

Random Dirichlet series arising from records

We study the distributions of the random Dirichlet series with parameters $(s, β)$ defined by $$ S=\sum_{n=1}^{\infty}\frac{I_n}{n^s}, $$ where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value $1$ with probability $1/n^β$ and value $0$ otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in extreme value theory in statistics. We show that when $s>0$ and $0< β\le 1$ with $s+β>1$ the distribution of $S$ has a density; otherwise it is purely atomic or not defined because of divergence. In particular, in the case when $s>0$ and $β=1$, we prove that for every $0 1$ it is unbounded. In the case when $s>0$ and $0<β<1$ with $s+β>1$, the density is smooth. To show the absolute continuity, we obtain estimates of the Fourier transforms, employing van der Corput's method to deal with number-theoretic problems. We also give further regularity results of the densities, and present an example of non atomic singular distribution which is induced by the series restricted to the primes.

math.PR↗
arXiv2015

Random walks colliding before getting trapped

Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $λ$ if it jumps at rate $λ$ according to the matrix $P$. Fix $λ_X,λ_Y,λ_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $λ_X,λ_Y$ and $λ_Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $λ_X,λ_Y$ and $λ_Z$ with $\max(λ_X,λ_Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $λ_X=λ_Y=1$ and $λ_Z=0$ we prove a strengthening of our main theorem using a martingale argument. We provide an example showing the transitivity assumption cannot be removed for general $λ_X,λ_Y$ and $λ_Z$.

math.PR↗
arXiv2015

Topics in Markov chains: mixing and escape rate

These are the notes for the minicourse on Markov chains delivered at the Saint Petersburg Summer School, June 2012. The main emphasis is on methods for estimating mixing times (for finite chains) and escape rates (for infinite chains). Lamplighter groups are key examples in both topics and the Varopolous-Carne long range estimate is useful in both settings.

math.PR↗
arXiv2015

Collisions of Random Walks in Reversible Random Graphs

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in $\mathbb{Z}^2$.

math.PR↗
arXiv2015

Heat diffusion with frozen boundary

Consider "Frozen Random Walk" on $\mathbb{Z}$: $n$ particles start at the origin. At any discrete time, the leftmost and rightmost $\lfloor{\frac{n}{4}}\rfloor$ particles are "frozen" and do not move. The rest of the particles in the "bulk" independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To this end we study the following deterministic mass splitting process: start with mass $1$ at the origin. At each step the extreme quarter mass on each side is "frozen". The remaining "free" mass in the center evolves according to the discrete heat equation. We establish diffusive behavior of this mass evolution and identify the scaling limit under the assumption of its existence. It is natural to expect the limit to be a truncated Gaussian. A naive guess for the truncation point might be the $1/4$ quantile points on either side of the origin. We show that this is not the case and it is in fact determined by the evolution of the second moment of the mass distribution.

math.PR↗
arXiv2015

Bandit Convex Optimization: sqrt{T} Regret in One Dimension

We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis features a novel use of convexity, formalized as a "local-to-global" property of convex functions, that may be of independent interest.

cs.LG↗
arXiv2015

Characterization of cutoff for reversible Markov chains

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 stationary measure at least $α$, for some $α\in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $α$ was not hit by time $t$. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the ratio of their relaxation-times and their (lazy) mixing-times tends to 0.

math.PR↗
arXiv2015

Formation of an interface by competitive erosion

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 kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability.

math.PR↗
arXiv2014

Intersection and mixing times for reversible chains

Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time is always upper bounded by the expected intersection time taken over the worst starting states. For random walks on trees we show the two quantities are equivalent. We obtain an expression for the expected intersection time in terms of the eigenvalues for reversible and transitive chains. For such chains we also show that it is up to constants the geometric mean of n and E[I], where I is the number of intersections up to the uniform mixing time. Finally for random walks on regular graphs we obtain sharp inequalities that relate the expected intersection time to maximum hitting time and mixing time.

math.PR↗
arXiv2014

Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > ε$ for some $ε> 0$ independent of $n$, even when an adversary is allowed to conjugate each of the four by a possibly different element of $§_n$ (in other words, the cycle types already guarantee generation of $\mathcal{S}_n$). This is closely related to the following random set model. A random set $M \subseteq \mathbb{Z}^+$ is generated by including each $n \geq 1$ independently with probability $1/n$. The sumset $\text{sumset}(M)$ is formed. Then at most four independent copies of $\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite.

math.PR↗
arXiv2014

Restrictions of Brownian motion

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $α>\frac12$ and $1\leq β\leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac12$ and $B\colon A\to\mathbb{R}$ is $α$-Hölder continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\fracβ{2}$ and $B\colon A\to\mathbb{R}$ has finite $β$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.

math.PR↗
arXiv2014

Rigidity and tolerance for perturbed lattices

A perturbed lattice is a point process $Π=\{x+Y_x:x\in \mathbb{Z}^d\}$ where the lattice points in $\mathbb{Z}^d$ are perturbed by i.i.d.\ random variables $\{Y_x\}_{x\in \mathbb{Z}^d}$. A random point process $Π$ is said to be rigid if $|Π\cap B_0(1)|$, the number of points in a ball, can be exactly determined given $Π\setminus B_0(1)$, the points outside the ball. The process $Π$ is called deletion tolerant if removing one point of $Π$ yields a process with distribution indistinguishable from that of $Π$. Suppose that $Y_x\sim N_d(0,σ^2 I)$ are Gaussian vectors with with $d$ independent components of variance $σ^2$. Holroyd and Soo showed that in dimensions $d=1,2$ the resulting Gaussian perturbed lattice $Π$ is rigid and deletion intolerant. We show that in dimension $d\geq 3$ there exists a critical parameter $σ_r(d)$ such that $Π$ is rigid if $σ<σ_r$ and deletion tolerant (hence non-rigid) if $σ>σ_r$.

math.PR↗
arXiv2014

The range of a rotor walk

In a \emph{rotor walk} the exits from each vertex follow a prescribed periodic sequence. On an infinite Eulerian graph embedded periodically in $\R^d$, we show that any simple rotor walk, regardless of rotor mechanism or initial rotor configuration, visits at least on the order of $t^{d/(d+1)}$ distinct sites in $t$ steps. We prove a shape theorem for the rotor walk on the comb graph with i.i.d.\ uniform initial rotors, showing that the range is of order $t^{2/3}$ and the asymptotic shape of the range is a diamond. Using a connection to the mirror model and critical percolation, we show that rotor walk with i.i.d.\ uniform initial rotors is recurrent on two different directed graphs obtained by orienting the edges of the square grid, the Manhattan lattice and the $F$-lattice. We end with a short discussion of the time it takes for rotor walk to cover a finite Eulerian graph.

math.PR↗
arXiv2014

Competing first passage percolation on random regular graphs

We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on $N$ vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy $Θ(1)N^α$ vertices, for some $0 < α< 1$. The value of $α$ is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on $d$ dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each.

math.PR↗
arXiv2014

Surprise probabilities in Markov chains

In a Markov chain started at a state $x$, the hitting time $τ(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(τ(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{n}{t}$. 2) In a reversible chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{\sqrt{2n}}{t}$ for $t \ge 4n + 4$. 3) For random walk on a simple graph with $n \ge 2$ vertices, $\mathbf{P}_x(τ(y) = t) \le \frac{4e \log n}{t}$. We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain. To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): For random walk on an $n$-vertex graph, for every initial vertex $x$, \[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \]

math.PR↗
arXiv2014

Separating signal from noise

Suppose that a sequence of numbers $x_n$ (a `signal') is transmitted through a noisy channel. The receiver observes a noisy version of the signal with additive random fluctuations, $x_n + ξ_n$, where $ξ_n$ is a sequence of independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted? In this paper we consider the case in which signals are infinite sequences and the recovery or detection are required to hold with probability one. We provide conditions on the signal space for checking whether detection or recovery are possible. We also analyze in detail several examples including spaces of Fourier transforms of measures, spaces with fixed amplitudes and the space of almost periodic functions. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible.

math.PR↗