arXiv subjects

Yuval Peres

Publications and source records attributed to Yuval Peres.

At least 199 records · Page 11Linked to original sources

arXiv2010

Finding Hidden Cliques in Linear Time with High Probability

We are given a graph $G$ with $n$ vertices, where a random subset of $k$ vertices has been made into a clique, and the remaining edges are chosen independently with probability $\tfrac12$. This random graph model is denoted $G(n,\tfrac12,k)$. The hidden clique problem is to design an algorithm that finds the $k$-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant $c > 0$. Recently, an algorithm that solves the same problem was proposed by Feige and Ron. It has the advantages of being simpler and more intuitive, and of an improved running time of $O(n^2)$. However, the analysis in the paper gives success probability of only $2/3$. In this paper we present a new algorithm for finding hidden cliques that both runs in time $O(n^2)$, and has a failure probability that is less than polynomially small.

math.CO↗

arXiv2010

Thick points of the Gaussian free field

Let $U\subseteq\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U\nabla f(x)\cdot \nabla g(x)\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/π}\log\frac{1}{r}$ as $r\to 0$. We show that for each $0\leq a\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $ν_{2-a}(T(a;U))=\infty$ when $0 2$. Furthermore, we prove that $T(a;U)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $γ$ given formally by $Γ(dz)=e^{\sqrt{2π}γF(z)}\,dz$ considered by Duplantier and Sheffield.

math.PR↗

arXiv2010

A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group $G$ and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in $Θ(\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on $J$ processors we find that $Θ(\sqrt{|G|}/J)$ steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from $G$.

math.PR↗

arXiv2010

The evolution of the cover time

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation. We refine this upper bound, and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large d. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman, is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.

math.PR↗

arXiv2010

Diameters in supercritical random graphs via first passage percolation

We study the diameter of $C_1$, the largest component of the Erdős-Rényi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+ε}n$ where $ε^3 n \to \infty$ and $ε=o(1)$. This parameter was extensively studied for fixed $ε> 0$, yet results for $ε=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $ε^3 n\to\infty$ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $1000/7$. We show that throughout the emerging supercritical phase, i.e. for any $ε=o(1)$ with $ε^3 n \to \infty$, the diameter of $C_1$ is with high probability asymptotic to $D(ε,n)=(3/ε)\log(ε^3 n)$. This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $(2/3)D(ε,n)$, and the maximal distance in $C_1$ between any pair of kernel vertices is w.h.p. asymptotic to $(5/9)D(ε,n)$.

math.CO↗

arXiv2010

Hitting times for random walks with restarts

The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $T(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.

math.PR↗

arXiv2010

Collisions of Random Walks

A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $\Z^d$ with $d \ge 19$ and the uniform spanning tree in $\Z^2$ all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.

math.PR↗

arXiv2010

Brownian motion with variable drift can be space-filling

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $α< 1/d$ we construct an $α$-Hölder continuous function $f \colon [0,1] \to \mathbb{R}^d$ so that the range of $B-f$ covers an open set. This strengthens a result of Graversen (1982) and answers a question of Le Gall (1988).

math.PR↗

arXiv2010

The critical Ising model on trees, concave recursions and nonlinear capacity

We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an $L^3$ capacity arises. In particular, on a spherically symmetric tree that has $n^αb^n$ vertices at level $n$ (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if $α\le1/2$. Our proofs are based on a new link between nonlinear recursions on trees and $L^p$ capacities.

math.PR↗

arXiv2010

New coins from old, smoothly

Given a (known) function $f:[0,1] \to (0,1)$, we consider the problem of simulating a coin with probability of heads $f(p)$ by tossing a coin with unknown heads probability $p$, as well as a fair coin, $N$ times each, where $N$ may be random. The work of Keane and O'Brien (1994) implies that such a simulation scheme with the probability $¶_p(N<\infty)$ equal to 1 exists iff $f$ is continuous. Nacu and Peres (2005) proved that $f$ is real analytic in an open set $S \subset (0,1)$ iff such a simulation scheme exists with the probability $¶_p(N>n)$ decaying exponentially in $n$ for every $p \in S$. We prove that for $α>0$ non-integer, $f$ is in the space $C^α[0,1]$ if and only if a simulation scheme as above exists with $¶_p(N>n) \le C (Δ_n(p))^α$, where $Δ_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}$. The key to the proof is a new result in approximation theory: Let $\B_n$ be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree $n$. We show that a function $f:[0,1] \to (0,1)$ is in $C^α[0,1]$ if and only if $f$ has a series representation $\sum_{n=1}^\infty F_n$ with $F_n \in \B_n$ and $\sum_{k>n} F_k(x) \le C(Δ_n(x))^α$ for all $ x \in [0,1]$ and $n \ge 1$. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some $ϕ_n \in \B_n$ satisfy $|f(x)-ϕ_n(x)| \le C (Δ_n(x))^α$ for all $ x \in [0,1]$ and $n \ge 1$, then $f \in C^α[0,1]$.

math.PR↗

arXiv2009

Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z^d. Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h <= 2d-2, the diameter of the set of sites that topple has order n^{1/d}. This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d-1. On the other hand, we show that if the background height 2d-2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).

math.CO↗

arXiv2009

Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators

Consider a Markov chain $(ξ_v)_{v \in V} \in [k]^V$ on the infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \geq 2$, $k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real second-largest (in absolute value) eigenvalue $λ$ with corresponding real eigenvector $ν\neq 0$. Letting $σ_v = ν_{ξ_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the "recontruction problem" on trees: \begin{equation*} S_n = (bλ)^{-n} \sum_{x\in L_n} σ_x. \end{equation*} As noted by Mossel and Peres, when $bλ^2 > 1$ (the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$ has uniformly bounded variance. Here, we give bounds on the moment-generating functions of $S_n$ and $S_n^2$ when $bλ^2 > 1$. Our results have implications for the inference of evolutionary trees.

math.PR↗

arXiv2009

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if U\subset\R^n is an open domain whose closure \overline{U} is compact in the path metric, and F is a Lipschitz function on \partial{U}, then for each β\in\R there exists a unique viscosity solution to the β-biased infinity Laplacian equation β|\nabla u| + Δ_\infty u=0 on U that extends F, where Δ_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased \eps-game as follows. The starting position is x_0 \in U. At the k^\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \exp(β\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in \partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps \to 0, where u is the unique extension of F to \overline{U} that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.

math.AP↗

arXiv2009

Anatomy of a young giant component in the random graph

We provide a complete description of the giant component of the Erdős-Rényi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+ε)/n$ where $ε^3 n \to \infty$ and $ε=o(1)$. Our description is particularly simple for $ε= o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 ε^3 n$ and variance $ε^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/ε$. Finally, attach an independent Poisson($1-ε$)-Galton-Watson tree to each vertex. A similar picture is obtained for larger $ε=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $ε^k n$. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of $ε$, as well as the mixing time of the random walk on $C_1$.

math.CO↗

arXiv2009

Convergence of Local Dynamics to Balanced Outcomes in Exchange Networks

Bargaining games on exchange networks have been studied by both economists and sociologists. A Balanced Outcome for such a game is an equilibrium concept that combines notions of stability and fairness. In a recent paper, Kleinberg and Tardos introduced balanced outcomes to the computer science community and provided a polynomial-time algorithm to compute the set of such outcomes. Their work left open a pertinent question: are there natural, local dynamics that converge quickly to a balanced outcome? In this paper, we provide a partial answer to this question by showing that simple edge-balancing dynamics converge to a balanced outcome whenever one exists.

cs.GT↗

arXiv2009

Heaviness in Toral Rotations

We investigate the dimension of the set of points in the d-torus which have the property that their orbit under rotation by some alpha hits a fixed closed target A more often than expected for all finite initial portions. An upper bound for the lower Minkowski dimension of this "strictly heavy set" H(A,alpha) is found in terms of the upper Minkowski dimension of the boundary of A, as well as k, the Diophantine approximability from below of the Lebesgue measure of A. The proof extends to translations in compact abelian groups more generally than just the torus, most notably the p-adic integers.

math.DS↗

arXiv2009

Convolutions of Cantor measures without resonance

Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0 1$ and $\log (1/3) /\log (1/4)$ is irrational.

math.CA↗

arXiv2009

A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes

In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.

math.DS↗