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Random complex zeroes, I. Asymptotic normality
, 2003
"... We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes. Introduction and the main result Zeroes of random p ..."
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Cited by 12 (3 self)
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We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes. Introduction and the main result Zeroes of random polynomials and other analytic functions were studied by mathematicians and physicists under various assumptions on random coefficients. One class of models introduced not long ago by Bogomolny, Bohigas and Leboeuf [5, 6], Kostlan [16], and Shub and Smale [23] has a remarkably
Brownian survival and Lifshitz tail in perturbed lattice disorder, arXiv:0807.2486
"... We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics of the logarithm of the survival probability up to ..."
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Cited by 2 (1 self)
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We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant. As applications, we show the Lifshitz tail effect of the density of states of the associated random Schrödinger operator and derive a quantitative estimate for the strength of intermittency in the parabolic Anderson problem.
Gravitational allocation to Poisson points
, 2008
"... For d ≥ 3, we construct a nonrandomized, fair and translationequivariant allocation of Lebesgue measure to the points of a standard Poisson point process in Rd, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force fiel ..."
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Cited by 1 (0 self)
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For d ≥ 3, we construct a nonrandomized, fair and translationequivariant allocation of Lebesgue measure to the points of a standard Poisson point process in Rd, 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 − cR(log R) αd for all R> 2, where: αd = d−2
Uniformly spread measures and vector fields
, 801
"... We show that two different ideas of uniform spreading of locally finite measures in the ddimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connec ..."
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Cited by 1 (0 self)
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We show that two different ideas of uniform spreading of locally finite measures in the ddimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure. 1
Estimates on the Probability of Outliers for Real Random BargmannFock functions.
, 807
"... In this paper we consider the distribution of the zeros of a real random BargmannFock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there ar ..."
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In this paper we consider the distribution of the zeros of a real random BargmannFock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there are no zeros in [−r, r] m ⊂ R m decays at least exponentially in terms of r m. For this event we also prove a lower bound on the order of decay, which we do not expect to be sharp. Secondly, we compute the order of decay for the probability of families of events where the volume of the complex zero set is either much larger or much smaller then expected. 1. Introduction. Random functions provide techniques to study typical properties of elements of a Hilbert space, and have been used to shed light on the zero set of elements of a Hilbert Space of functions. We will define real random BargmannFock functions as a linear combination of basis functions, where
Author manuscript, published in "48th Annual Allerton Conference (2010)" DOI: 10.1109/ALLERTON.2010.5707086 Connectivity in SubPoisson Networks
, 2010
"... Abstract: We consider a class of point processes, which we call subPoisson; these are point processes that can be directionallyconvexly (dcx) dominated by some Poisson point process. The dcx order has already been shown in [4] useful in comparing various point process characteristics, including Rip ..."
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Abstract: We consider a class of point processes, which we call subPoisson; these are point processes that can be directionallyconvexly (dcx) dominated by some Poisson point process. The dcx order has already been shown in [4] useful in comparing various point process characteristics, including Ripley’s and correlation functions as well as shotnoise fields generated by point processes, indicating in particular that smaller in the dcx order processes exhibit more regularity (less clustering, less voids) in the repartition of their points. Using these results, in this paper we study the impact of the dcx ordering of point processes on the properties of two continuum percolation models, which have been proposed in the literature to address macroscopic connectivity properties of large wireless networks. As the first main result of this paper, we extend the classical result on the existence of phase transition in the percolation of the Gilbert’s graph (called also the Boolean model), generated by a homogeneous Poisson point process, to the class of homogeneous subPoisson processes. We also extend a recent result of the same nature for the SINR graph, to subPoisson point processes. Finally, we show examples the socalled perturbed lattices, which are subPoisson. More generally, perturbed lattices provide some spectrum of models that ranges from periodic grids, usually considered in cellular network context, to Poisson adhoc networks, and to various more clustered point processes including some doubly stochastic Poisson ones. Index Terms—percolation, dcx order, Gilbert’s graph, Boolean