arXiv subjects

Ron Peled

Publications and source records attributed to Ron Peled.

67 records · Page 4Linked to original sources

arXiv2012

Lipschitz Functions on Expanders are Typically Flat

This work studies the typical behavior of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions that change by at most M along edges) and integer-homomorphisms (functions that change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability to take other values.

math.PR↗

arXiv2012

On K-wise Independent Distributions and Boolean Functions

We pursue a systematic study of the following problem. Let f:{0,1}^n -> {0,1} be a (usually monotone) Boolean function whose behaviour is well understood when the input bits are identically independently distributed. What can be said about the behaviour of the function when the input bits are not completely independent, but only k-wise independent, i.e. every subset of k bits is independent? more precisely, how high should k be so that any k-wise independent distribution "fools" the function, i.e. causes it to behave nearly the same as when the bits are completely independent? We analyze several well known Boolean functions (including AND, Majority, Tribes and Percolation among others), some of which turn out to have surprising properties. In some of our results we use tools from the theory of the classical moment problem, seemingly for the first time in this subject, to shed light on these questions.

math.PR↗

arXiv2011

Probabilistic existence of rigid combinatorial structures

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

math.CO↗

arXiv2011

The Maximal Probability that k-wise Independent Bits are All 1

A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? The question is a type of discrete moment problem, and our approach is based on showing that bounds obtained from the theory of the classical moment problem provide good approximations for it. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros.

math.PR↗

arXiv2011

Odd cutsets and the hard-core model on Z^d

We consider the hard-core lattice gas model on Z^d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds Cd^{-1/3}(log d)^2, the model exhibits multiple hard-core measures, thus improving the previous bound of Cd^{-1/4}(log d)^{3/4} given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in Z^d, the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial analysis of the structure of these cutsets yielding a quantitative form of concentration for their possible shapes as the dimension d tends to infinity. This analysis relies upon and improves previous results obtained by the first author.

math.PR↗

arXiv2010

On rough isometries of Poisson processes on the line

Intuitively, two metric spaces are rough isometric (or quasi-isometric) if their large-scale metric structure is the same, ignoring fine details. This concept has proven fundamental in the geometric study of groups. Abért, and later Szegedy and Benjamini, have posed several probabilistic questions concerning this concept. In this article, we consider one of the simplest of these: are two independent Poisson point processes on the line rough isometric almost surely? Szegedy conjectured that the answer is positive. Benjamini proposed to consider a quantitative version which roughly states the following: given two independent percolations on $\mathbb {N}$, for which constants are the first $n$ points of the first percolation rough isometric to an initial segment of the second, with the first point mapping to the first point and with probability uniformly bounded from below? We prove that the original question is equivalent to proving that absolute constants are possible in this quantitative version. We then make some progress toward the conjecture by showing that constants of order $\sqrt{\log n}$ suffice in the quantitative version. This is the first result to improve upon the trivial construction which has constants of order $\log n$. Furthermore, the rough isometry we construct is (weakly) monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them.

math.PR↗

arXiv2010

Simple Universal Bounds for Chebyshev-Type Quadratures

A Chebyshev-type quadrature for a probability measure sigma is a distribution which is uniform on n points and has the same first k moments as sigma. We give an upper bound for the minimal n required to achieve a given degree k, for sigma supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of sigma and is applicable in wide generality. We also obtain a lower bound for the required number of nodes which only uses estimates on the moments of sigma. Examples illustrating the sharpness of our bounds are given. As a corollary of our results, we obtain an apparently new result on the Gaussian quadrature. In addition, we suggest another approach to bounding the minimal number of nodes required in a Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose the challenge of analyzing its performance. A preliminary result in this direction is proved for the uniform measure on the cube. Finally, we apply our bounds to the construction of point sets on the sphere and cylinder which form local approximate Chebyshev-type quadratures. These results were needed recently in the context of understanding how well can a Poisson process approximate certain continuous distributions. The paper concludes with a list of open questions.

math.CA↗

arXiv2009

Stationary map coloring

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.

math.PR↗

arXiv2009

Phase Transitions in Gravitational Allocation

Given a Poisson point process of unit masses (``stars'') in dimension d>=3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the diameters of these cells have exponential tails. Here we analyze the quantitative geometry of the cells and show that their large deviations occur at the stretched-exponential scale. More precisely, the probability that mass exp(-R^gamma) in a cell travels distance R decays like exp(-R^f_d(gamma)) where we identify the functions f_d exactly. These functions are piecewise smooth and the discontinuities of f_d' represent phase transitions. In dimension d=3, the large deviation is due to a ``distant attracting galaxy'' but a phase transition occurs when f_3(gamma)=1 (at that point, the fluctuations due to individual stars dominate). When d>=5, the large deviation is due to a thin tube (a ``wormhole'') along which the star density increases monotonically, until the point f_d(gamma)=1 (where again fluctuations due to individual stars dominate). In dimension 4 we find a double phase transition, where the transition between low-dimensional behavior (attracting galaxy) and high-dimensional behavior (wormhole) occurs at gamma=4/3. As consequences, we determine the tail behavior of the distance from a star to a uniform point in its cell, and prove a sharp lower bound for the tail probability of the cell's diameter, matching our earlier upper bound.

math.PR↗

arXiv2008

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator $f\mapsto{1/2}f''$. We then consider the special case where the state space is the self-similar set $\{\pm q^k:k\in \mathbb{Z}\}\cup\{0\}$ for some $q>1$. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's ``lost'' notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, $q$-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local time -- both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve $q$-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

math.PR↗

arXiv2008

Growth of the Number of Spanning Trees of the Erdös-Rényi Giant Component

The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\PGW(λ)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $λ$. Suppose that $λ^*>λ>1$. We show that $\PGW(λ^*)$ conditioned to survive forever stochastically dominates $\PGW(λ)$ conditioned to survive forever.

math.PR↗

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↗

arXiv2004

Restoring Topology from Shifts

It is known that the topology of a Polish group is uniquely determined by its Borel structure and group operations, but this does not give us a way to find the topology. In this article we expand on this theorem and give a criterion for a measurable function on the group to be continuous. We show that it is continuous iff there exists some second countable topology in which all the shifts of the function are continuous. Here measurability can be taken to be measurable with respect to Haar measure (in locally compact groups), or having the Baire property or universally measurable (in Abelian Polish groups). Our results appear to be new even when the group is $\R$. As a special case we get that a measurable homomorphism is continuous (a known result). As an application, We give a first proof of a dichotomy guessed by Tsirelson long ago for stationary stochastic processes. Either the process is sample continuous, or its paths cannot be continuous in any second countable topology on any non-null set.

math.GN↗