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Yuval Peres

Publications and source records attributed to Yuval Peres.

At least 235 records · Page 13Linked to original sources
arXiv2008

Poisson Matching

Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.

math.PR↗
arXiv2008

The power law for the Buffon needle probability of the four-corner Cantor set

Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres and Solomyak. ($\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). We obtain a power law bound by combining analytic and combinatorial ideas.

math.CA↗
arXiv2007

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).

math.PR↗
arXiv2007

What is the probability of intersecting the set of Brownian double points?

We give potential theoretic estimates for the probability that a set $A$ contains a double point of planar Brownian motion run for unit time. Unlike the probability for $A$ to intersect the range of a Markov process, this cannot be estimated by a capacity of the set $A$. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously. We also give a polar decomposition of $A$ into a set that never intersects the set of Brownian double points and a set for which intersection with the set of Brownian double points is the same as intersection with the Brownian path.

math.PR↗
arXiv2007

The critical random graph, with martingales

We give a short proof that the largest component of the random graph $G(n, 1/n)$ is of size approximately $n^{2/3}$. The proof gives explicit bounds for the probability that the ratio is very large or very small.

math.PR↗
arXiv2007

Embeddings of discrete groups and the speed of random walks

For a finitely generated group G and a banach space X let α^*_X(G) (respectively α^#_X(G)) be the supremum over all α\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α^*(G)=α^*_{L_2}(G) (respectively α^#(G)= α_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then α^#_X(G)\le \frac{1}{pβ^*(G)}. Here β^*(G) is the largest β\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^β, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if α^*(G)\ge 1/2 then α^*(G\bwr \Z)\ge \frac{2α^*(G)}{2α^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then α^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.

math.MG↗
arXiv2007

The wreath product of Z with Z has Hilbert compression exponent 2/3

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $α\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\|f(x)-f(y)\|_2\ge cd(x,y)^α.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

math.MG↗
arXiv2007

How large a disc is covered by a random walk in n steps?

We show that the largest disc covered by a simple random walk (SRW) on $\mathbb{Z}^2$ after n steps has radius n^{1/4+o(1)}, thus resolving an open problem of Révész [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed $\ell$, the largest disc completely covered at least $\ell$ times by the SRW also has radius n^{1/4+o(1)}. However, the largest disc completely covered by each of $\ell$ independent simple random walks on $\mathbb{Z}^2$ after $n$ steps is only of radius $n^{1/(2+2\sqrt{\ell})+o(1)}$. We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction $α$ of the maximum number of visits to any site during the first $n$ steps of the SRW on $\mathbb{Z}^2$, is $n^{(1-\sqrtα)/4+o(1)}$. We also show that almost surely, for infinitely many values of $n$ it takes about $n^{1/2+o(1)}$ steps after step n for the SRW to reach the first previously unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised by Révész [Ann. Probab. 21 (1993) 318--328].

math.PR↗
arXiv2007

Critical percolation on random regular graphs

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In particular, we show that there is a scaling window of width n^{-1/3} around p_c in which the sizes of the largest components are roughly n^{2/3} and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps(n)n^{1/3} tends to infinity, the sizes of the largest components are concentrated around an explicit function of n and eps(n) which is of order o(n^{2/3}). In the supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but eps(n)n^{1/3} tends to infinity, the size of the largest component is concentrated around the value (2d/(d-2))\eps(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.

math.PR↗
arXiv2007

Maximum overhang

How far can a stack of $n$ identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order $\log n$. Recently, Paterson and Zwick constructed $n$-block stacks with overhangs of order $n^{1/3}$, exponentially better than previously thought possible. We show here that order $n^{1/3}$ is indeed best possible, resolving the long-standing overhang problem up to a constant factor.

math.HO↗
arXiv2007

Trees and Markov convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors.

math.MG↗
arXiv2007

Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation

Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $δ(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lovász local lemma to establish that $δ(x)>cr|\log r|^{-1}$ for some $x$, which implies that the chromatic number of $G$ is at most $Cr^{-1} |\log r|$. This is sharp up to the logarithmic factor.

math.CO↗
arXiv2007

Gravitational allocation to Poisson points

For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R).

math.PR↗
arXiv2006

Minimal spanning forests

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In $\mathbb{Z}^d$ this was proved by Alexander [Ann. Probab. 23 (1995) 87--104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are ``thin'' in a different sense, namely, on any graph, each component tree in the WMSF has $p_{\mathrm{c}}=1$ a.s., where $p_{\mathrm{c}}$ denotes the critical probability for having an infinite cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be ``thick'': on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We also show that the number of infinite clusters for Bernoulli($p_{\mathrm{u}}$) percolation is at most the number of components of the FMSF, where $p_{\mathrm{u}}$ denotes the critical probability for having a unique infinite cluster. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.

math.PR↗
arXiv2006

A stable marriage of Poisson and Lebesgue

Let $Ξ$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $Ξ$ centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into polyhedral regions (of varying sizes) by allocating each site of ${\mathbb{R}}^d$ to the closest center. Here we study ``fair'' allocations of ${\mathbb{R}}^d$ to $Ξ$ in which the regions allocated to different centers have equal volumes. We prove that if $Ξ$ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center $ξ$ is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than $ξ$. We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in $d>1$.

math.PR↗
arXiv2006

A Central Limit Theorem for biased random walks on Galton-Watson trees

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $λ$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $λ/(λ+d_v)$, and moves to each of the offspring with probability $1/(λ+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $ 0<λ m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $λ=m$, where the CLT has the following form: for almost every ${\cal T}$, the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for $λ=1$) and the construction of appropriate harmonic coordinates.

math.PR↗
arXiv2006

Determinantal Processes and Independence

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

math.PR↗