rigidity and tolerance in point processes gaussian zeros and ginibre eigenvalues let be a translation invariant point process on the complex plane c and let d c be a bounded open set we ask the following what does the point configuration out obtained by taking the points of outside d tell us about the point configuration in of inside d we show that for the ginibre ensemble out determines the number of points in in for the translation invariant zero process of a planar gaussian analytic function we show that out determines the number as well as the center of mass of the points in in further in both models we prove that the outside says nothing more about the inside in the sense that the conditional distribution of the inside points given the outside is mutually absolutely continuous with respect to the lebesgue measure on its supporting submanifold

outliers in the spectrum of iid matrices with bounded rank perturbations it is known that if one perturbs a large iid random matrix by a bounded rank error then the majority of the eigenvalues will remain distributed according to the circular law however the bounded rank perturbation may also create one or more outlier eigenvalues we show that if the perturbation is small then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation but if the perturbation is large then many more outliers can be created and their law is governed by the zeroes of a random laurent series with gaussian coefficients on the other hand these outliers may be eliminated by enforcing a row sum condition on the final matrix

from random matrices to random analytic functions we consider two families of random matrix valued analytic functions 1 g 1 zg 2 and 2 g 0 zg 1 z 2 g 2 where g i are n x n random matrices with independent standard complex gaussian entries the random set of z where these matrix analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane respectively the kernels of these determinantal processes are reproducing kernels of certain hilbert spaces bargmann fock spaces of holomorphic functions on the corresponding surfaces along with the new results this also gives a unified framework in which to view a theorem of peres and virag n 1 in the second setting above and a well known result of ginibre on gaussian random matrices which may be viewed as an analogue of our results in the whole plane

random weighted shifts abstract in this paper we initiate the study of a fundamental yet untapped random model of non selfadjoint bounded linear operators acting on a separable complex hilbert space we replace the weights w n 1 in the classical unilateral shift t defined as t e n w n e n 1 where e n n 1 form an orthonormal basis of a complex hilbert space by a sequence of i i d random variables x n n 1 that is w n x n this paper answers basic questions concerning such a model we propose that this model can be studied in comparison with the classical hardy bergman dirichlet spaces in function theoretic operator theory we calculate the spectra and determine their fine structures section 3 we classify the samples up to four equivalence relationships section 4 we introduce a family of random hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces section 5 we compare them with three types of classical operators section 6 this is achieved in the form of generalized von neumann inequalities the invariant subspaces are shown to admit arbitrarily large indices and their semi invariant subspaces model arbitrary contractions almost surely we discuss a beurling type theorem section 7 we determine various non selfadjoint algebras generated by t section 8 their dynamical properties are clarified section 9 their iterated aluthge transforms are shown to converge section 10 in summary they provide a new random model from the viewpoint of probability theory and they provide a new class of analytic functional hilbert spaces from the viewpoint of operator theory the technical novelty in this paper is that the methodology used draws from three largely separate sources probability theory functional hilbert spaces and the approximation theory of bounded operators

the z measures on partitions pfaffian point processes and the matrix hypergeometric kernel abstract we consider a point process on one dimensional lattice originated from the harmonic analysis on the infinite symmetric group and defined by the z measures with the deformation jack parameter 2 we derive an exact pfaffian formula for the correlation function of this process namely we prove that the correlation function is given as a pfaffian with a 2 2 matrix kernel the kernel is given in terms of the gauss hypergeometric functions and can be considered as a matrix analogue of the hypergeometric kernel introduced by a borodin and g olshanski 2000 5 our result holds for all values of admissible complex parameters

determinantal point processes associated with hilbert spaces of holomorphic functions we study determinantal point processes on mathbb c induced by the reproducing kernels of generalized fock spaces as well as those on the unit disc mathbb d induced by the reproducing kernels of generalized bergman spaces in the first case we show that all reduced palm measures of the same order are equivalent the radon nikodym derivatives are computed explicitly using regularized multiplicative functionals we also show that these determinantal point processes are rigid in the sense of ghosh and peres hence reduced palm measures of different orders are singular in the second case we show that all reduced palm measures of all orders are equivalent the radon nikodym derivatives are computed using regularized multiplicative functionals associated with certain blaschke products the quasi invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

on the spectra of gaussian matrices abstract we give a simple characterization of the moduli of the eigenvalues of a complex gaussian matrix in terms of x 2 distributions we also show that the spectral radius of a k k complex gaussian matrix is stochastically smaller than the norm of a k k 1 real gaussian matrix

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

a generalisation of the relation between zeros of the complex kac polynomial and eigenvalues of truncated unitary matrices the zeros of the random laurent series 1 mu sum j 1 infty c j z j where each c j is an independent standard complex gaussian is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex gaussian matrix for the corresponding random maclaurin series obtained by the replacement z mapsto 1 z we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix since the correlation functions of the latter are known by taking an appropriate limit the correlation functions for the random maclaurin series can be determined only for mu rightarrow infty is a determinantal point process obtained for the one and two point correlations by regarding the maclaurin series as the limit of a random polynomial a direct calculation can also be given

equivalence of palm measures for determinantal point processes governed by bergman kernels for a determinantal point process induced by the reproducing kernel of the weighted bergman space a 2 u omega over a domain u subset mathbb c d we establish the mutual absolute continuity of reduced palm measures of any order provided that the domain u contains a non constant bounded holomorphic function the result holds in all dimensions the argument uses the h infty u module structure of a 2 u omega a corollary is the quasi invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of u

hole probability for zeroes of gaussian taylor series with finite radii of convergence we study a family of random taylor series begin aligned f z sum n ge 0 zeta n a n z n end aligned with radius of convergence almost surely 1 and independent identically distributed complex gaussian coefficients zeta 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 leqslant 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

outlier eigenvalues for deformed i i d random matrices we consider a square random matrix of size n of the form a y where a is deterministic and y has iid entries with variance 1 n under mild assumptions as n grows the empirical distribution of the eigenvalues of a y converges weakly to a limit probability measure beta on the complex plane this work is devoted to the study of the outlier eigenvalues i e eigenvalues in the complement of the support of beta even in the simplest cases a variety of interesting phenomena can occur as in earlier works we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of y or the jordan decomposition of a we also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a gaussian analytic function

su 1 1 random polynomials we study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries su 1 1 by utilizing both analytical and numerical techniques we first show that zeros of the su 1 1 random polynomial of degree n are concentrated in a narrow annulus of the order of n 1 around the unit circle on the complex plane and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit n our results are supported through various numerical simulations we then extend results of hannay 1 and bleher et al 2 to derive different formulae for correlations between zeros of the su 1 1 random analytic functions by applying the generalized kac rice formula we express the correlation functions in terms of some gaussian integrals which can be evaluated combinatorially as a finite sum over feynman diagrams or as a supersymmetric integral due to the su 1 1 symmetry the correlation functions depend only on the hyperbolic distances between the points on the unit disk and we obtain an explicit formula for the two point correlation function it displays quadratic repulsion at small distances and fast decay of correlations at infinity in an appendix to the paper we evaluate correlations between the outer zeros zj 1 of the su 1 1 random polynomial and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity

gaussian fluctuations for non hermitian random matrix ensembles consider an ensemble of n n non hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean square one if the entry distributions also possess bounded densities and finite 4 moments then z d bai ann probab 25 1997 494 529 has shown the ensemble to satisfy the circular law after scaling by a factor of 1 sqrt n and letting n the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane in this note we investigate fluctuations from the circular law in a more restrictive class of non hermitian matrices for which higher moments of the entries obey a growth condition the main result is a central limit theorem for linear statistics of type xn f k 1nf k where 1 2 n denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain the proof is inspired by bai and silverstein ann probab 32 2004 533 605 where the analogous result for random sample covariance matrices is established

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

overcrowding estimates for zeroes of planar and hyperbolic gaussian analytic functions we consider the point process of zeroes of certain gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r as r rightarrow infty for the planar gaussian analytic function sum n geq 0 frac a n z n sqrt n we show that this probability is asymptotic to e frac 1 2 m 2 log m for the hyperbolic gaussian analytic functions sum n geq 0 rho choose n 1 2 a n z n rho 0 we show that this probability decays like e cm 2

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

on adding a list of numbers and other one dependent determinantal processes adding a column of numbers produces carries along the way we show that random digits produce a pattern of carries with a neat probabilistic description the carries form a one dependent determinantal point process this makes it easy to answer natural questions how many carries are typical where are they located we show that many further examples from combinatorics algebra and group theory have essentially the same neat formulae and that any one dependent point process on the integers is determinantal the examples give a gentle introduction to the emerging fields of one dependent and determinantal point processes

random polynomials and approximate zeros of newton s method in this paper the authors study the size of the set of approximate zeros for newton s method for a randomly chosen polynomial over certain distributions for a degree d monic polynomial with coefficients chosen uniformly and independently in the unit ball the results of this paper show for example that the set of approximate zeros is at least cd 1 5 varepsilon for any positive varepsilon with c depending only on varepsilon

universality and scaling of correlations between zeros on complex manifolds we study the limit as n of the correlations between simultaneous zeros of random sections of the powers l n of a positive holomorphic line bundle l over a compact complex manifold m when distances are rescaled so that the average density of zeros is independent of n we show that the limit correlation is independent of the line bundle and depends only on the dimension of m and the codimension of the zero sets we also provide some explicit formulas for pair correlations in particular we prove that hannay s limit pair correlation function for su 2 polynomials holds for all compact riemann surfaces

large deviations for the fermion point process associated with the exponential kernel for the fermion point process on the whole complex plane associated with the exponential kernel e z bar w we show the central limit theorem for the random variable d r the number of points inside the ball d r of radius r as r and we establish the large deviation principle for the random variables r 2 d r r 0

a semiparametric mixture approach to case control studies with errors in covariables abstract methods are devised for estimating the parameters of a prospective logistic model in a case control study with dichotomous response d that depends on a covariate x for a portion of the sample both the gold standard x and a surrogate covariate w are available however for the greater portion of the data only the surrogate covariate w is available by using a mixture model the relationship between the true covariate and the response can be modeled appropriately for both types of data the likelihood depends on the marginal distribution of x and the measurement error density w x d the latter is modeled parametrically based on the validation sample the marginal distribution of the true covariate is modeled using a nonparametric mixture distribution in this way we can improve the efficiency and reduce the bias of the parameter estimates the results also apply when there is no validation data provided the error distribution is known or estimated from an independent data source many of the

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

zeros of the i i d gaussian power series a conformally invariant determinantal process consider the zero set of the random power series f z p anz n with i i d complex gaussian coefficientsan we show that these zeros form a determinantal process more precisely their joint intensity can be written as a minor of the bergman kernel we show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent 0 1 valued random variables xk where p xk 1 r 2k moreover the set of absolute values of the zeros of f has the

gap probabilities for the cardinal sine we study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the paley wiener space as a model case we consider real valued gaussian coefficients it is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length c 2012 elsevier inc all rights reserved

statistical properties of determinantal point processes in high dimensional euclidean spaces the goal of this paper is to quantitatively describe some statistical properties of higher dimensional determinantal point processes with a primary focus on the nearest neighbor distribution functions toward this end we express these functions as determinants of n n matrices and then extrapolate to n this formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in euclidean spaces of dimension d we also implement an algorithm due to hough et al for generating configurations of determinantal point processes in arbitrary euclidean spaces and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the fermi sphere point process for d 1 4 this homogeneous isotropic determinantal point process discussed also in a companion paper s torquato a scardicchio and c e zachary j stat mech theory exp 2008 p11019 i s the high dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble in addition to the nearest neighbor probability distribution we are able to calculate voronoi cells and nearest neighbor extrema statistics for the fermi sphere point process and we discuss these properties as the dimension d is varied the results in this paper accompany and complement analytical properties of higher dimensional determinantal point processes developed in a prior paper

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

skew orthogonal polynomials and the partly symmetric real ginibre ensemble the partly symmetric real ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti symmetric gaussian random matrices such matrices typically have both real and complex eigenvalues for a fixed number of real eigenvalues an earlier work has given the explicit form of the joint eigenvalue probability density function we use this to derive a pfaffian formula for the corresponding summed up generalized partition function this pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries it can be used too to give the explicit form of the correlation functions provided certain skew orthogonal polynomials are computed this task is accomplished in terms of hermite polynomials and allows us to proceed to analyze various scaling limits of the correlations including that in which the matrices are only weakly non symmetric