arXiv subjects

Ron Peled

Publications and source records attributed to Ron Peled.

At least 55 records · Page 3Linked to original sources

arXiv2017

Probabilistic existence of regular combinatorial structures

We show the existence of regular 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 structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

math.CO↗

arXiv2017

Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} ζ_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(ζ_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients $(a_n)$ display power-law behavior.

math.CV↗

arXiv2017

High-Dimensional Lipschitz Functions are Typically Flat

A homomorphism height function on the $d$-dimensional torus $\mathbb{Z}_n^d$ is a function taking integer values on the vertices of the torus with consecutive integers assigned to adjacent vertices. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. In each model, we consider the uniform distribution over such functions, subject to boundary conditions. We prove that in high dimensions, with zero boundary values, a typical function is very flat, having bounded variance at any fixed vertex and taking at most $C(\log n)^{1/d}$ values with high probability. Our results extend to any dimension $d\ge 2$, if $\mathbb{Z}_n^d$ is replaced by an enhanced version of it, the torus $\mathbb{Z}_n^d\times\mathbb{Z}_2^{d_0}$ for some fixed $d_0$. This establishes one side of a conjectured roughening transition in $2$ dimensions. The full transition is established for a class of tori with non-equal side lengths. We also find that when $d$ is taken to infinity while $n$ remains fixed, a typical function takes at most $r$ values with high probability, where $r=5$ for the homomorphism model and $r=4$ for the Lipschitz model. Suitable generalizations are obtained when $n$ grows with $d$. Our results apply also to the related model of uniform 3-coloring and establish, for certain boundary conditions, that a uniformly sampled proper 3-coloring of $\mathbb{Z}_n^d$ will be nearly constant on either the even or odd sub-lattice. Our proofs are based on a combinatorial transformation and on a careful analysis of the properties of a class of cutsets which we term odd cutsets. For the Lipschitz model, our results rely also on a bijection of Yadin. This work generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini, Yadin and Yehudayoff and answers a question of Benjamini, Häggström and Mossel.

math-ph↗

arXiv2017

Rigidity of 3-colorings of the discrete torus

We prove that a uniformly chosen proper $3$-coloring of the $d$-dimensional discrete torus has a very rigid structure when the dimension $d$ is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the $3$-state anti-ferromagnetic Potts model from statistical physics. Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain topological obstructions which appear when no boundary conditions are imposed on the model. Locally, a proper $3$-coloring defines the discrete gradient of an integer-valued height function which changes by exactly one between adjacent sites. However, these locally-defined functions do not always yield a height function on the entire torus, as the gradients may accumulate to a non-zero quantity when winding around the torus. Our main result is that in high dimensions, a global height function is well defined with high probability, allowing to deduce the rigid structure of the coloring from previously known results. Moreover, the probability that the gradients accumulate to a vector $m$, corresponding to the winding in each of the $d$ directions, is at most exponentially small in the product of $\|m\|_\infty$ and the area of a cross-section of the torus. In the course of the proof we develop discrete analogues of notions from algebraic topology. This theory is developed in some generality and may be of use in the study of other models.

math-ph↗

arXiv2017

Matrix regularizing effects of Gaussian perturbations

The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of $H$ in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of $H^{-1}$ and for the distribution of the norm of $H^{-1}$ applied to a fixed vector. The bounds are uniform in $A$ and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.

math.PR↗

arXiv2016

Exponential decay of loop lengths in the loop $O(n)$ model with large $n$

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

math-ph↗

arXiv2016

On the Wegner orbital model

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

math-ph↗

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

A differential version of the Chebyshev-Markov-Stieltjes inequalities

We show that a differential version of the classical Chebyshev-Markov-Stieltjes inequalities holds for a broad family of weight functions. Such a differential version appears to be new. Our results apply to weight functions which are bounded away from zero and piecewise absolutely continuous and yield effective estimates when the weight satisfies additional regularity conditions.

math.CA↗

arXiv2015

Delocalization of two-dimensional random surfaces with hard-core constraints

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. No convexity assumption is made and we include the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint. Our main result is that these surfaces delocalize, having fluctuations whose variance is at least of order $\log n$, where $n$ is the side length of the torus. We also show that the expected maximum of such surfaces is of order at least $\log n$. The main tool in our analysis is an adaptation to the lattice setting of an algorithm of Richthammer, who developed a variant of a Mermin-Wagner-type argument applicable to hard-core constraints. We rely also on the reflection positivity of the random surface model. The result answers a question mentioned by Brascamp, Lieb and Lebowitz 1975 on the hammock potential and a question of Velenik 2006.

math.PR↗

arXiv2015

Double roots of random Littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and asymptotic to $\frac{8\sqrt{3}}{πn^2}$ otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on { -1, 0, 1} and whose largest atom is strictly less than 1/\sqrt{3}. In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^{-2}) factor and we find the asymptotics of the latter probability.

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↗

arXiv2014

Bijective combinatorial proof of the commutation of transfer matrices in the dense O(1) loop model

The dense O(1) loop model is a statistical physics model with connections to the quantum XXZ spin chain, alternating sign matrices, the six-vertex model and critical bond percolation on the square lattice. When cylindrical boundary conditions are imposed, the model possesses a commuting family of transfer matrices. The original proof of the commutation property is algebraic and is based on the Yang-Baxter equation. In this paper we give a new proof of this fact using a direct combinatorial bijection.

math-ph↗

arXiv2014

Lengths of Monotone Subsequences in a Mallows Permutation

We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that q<1, our results apply when q is a function of n satisfying n(1-q) -> infty. The case that n(1-q)=O(1) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order n(1-q)^(1/2), whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q. We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order 1/(1-q). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.

math.PR↗

arXiv2013

Random Walk with Long-Range Constraints

We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly sampled from all graph homomorphisms from the graph P_{n,d} to the integers Z, where the graph P_{n,d} is the discrete segment {0,1,..., n} with edges between vertices of different parity whose distance is at most 2d+1. Such a graph homomorphism can be viewed as a height function whose values change by exactly one along edges of the graph P_{n,d}. We also consider a similarly defined model on the discrete torus. Benjamini, Yadin and Yehudayoff conjectured that this model undergoes a phase transition from a delocalized to a localized phase when d grows beyond a threshold c*log(n). We establish this conjecture with the precise threshold log_2(n). Our results provide information on the typical range and variance of the height function for every given pair of n and d, including the critical case when d-log_2(n) tends to a constant. In addition, we identify the local limit of the model, when d is constant and n tends to infinity, as an explicitly defined Markov chain.

math.PR↗

arXiv2013

Grounded Lipschitz functions on trees are typically flat

A grounded M-Lipschitz function on a rooted d-ary tree is an integer-valued map on the vertices that changes by at most along edges and attains the value zero on the leaves. We study the behavior of such functions, specifically, their typical value at the root v_0 of the tree. We prove that the probability that the value of a uniformly chosen random function at v_0 is more than M+t is doubly-exponentially small in t. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.

math.PR↗

arXiv2013

Poisson Thickening

Let X be a Poisson point process of intensity lambda on the real line. A thickening of it is a (deterministic) measurable function f such that the union of X and f(X) is a Poisson point process of intensity lambda' where lambda'>lambda. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo. We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.

math.PR↗

arXiv2012

A recursive construction of t-wise uniform permutations

We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t-wise uniform set of permutations on n objects whose size is at most t^2n, the first non-trivial construction of an infinite family of t-wise uniform sets for t \geq 4. If a non-trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction.

math.CO↗