persistence of gaussian processes non summable correlations suppose the auto correlations of real valued centered gaussian process z cdot are non negative and decay as rho s t for some rho cdot regularly varying at infinity of order alpha in 1 0 with i rho t int 0 t rho s ds its primitive we show that the persistence probabilities decay rate of log mathbb p sup t in 0 t z t 0 is precisely of order t i rho t log i rho t thereby closing the gap between the lower and upper bounds of newell and rosenblatt ann math stat 33 1306 1313 1962 which stood as such for over fifty years we demonstrate its usefulness by sharpening recent results of sakagawa adv appl probab 47 146 163 2015 about the dependence on d of such persistence decay for the langevin dynamics of certain nabla phi interface models on mathbb z d

random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

rotation to simple loadings using component loss functions the oblique case component loss functions clfs similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis it is shown how the shape of the clf affects the performance of the criterion it defines for example it is shown that monotone concave clfs give criteria that are minimized by loadings with perfect simple structure when such loadings exist moreover if the clfs are strictly concave minimizing must produce perfect simple structure whenever it exists examples show that methods defined by concave clfs perform well much more generally while it appears important to use a concave clf the specific clf used is less important for example the very simple linear clf gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods

simplimax oblique rotation to an optimal target with simple structure factor analysis and principal component analysis are usually followed by simple structure rotations of the loadings these rotations optimize a certain criterion e g varimax oblimin designed to measure the degree of simple structure of the pattern matrix simple structure can be considered optimal if a usually large number of pattern elements is exactly zero in the present paper a class of oblique rotation procedures is proposed to rotate a pattern matrix such that it optimally resembles a matrix which has an exact simple pattern it is demonstrated that this method can recover relatively complex simple structures where other well known simple structure rotation techniques fail

persistence probability of random weyl polynomial in this paper we obtain the persistence exponents of random weyl polynomials in both cases the nonnegative axis and the whole real axis our result confirms the predictions given by schehr and majumdar j stat phys 132 2 235 273 2008 in the nonnegative axis case dembo and mukherjee ann probab 43 1 85 118 2015 gave an upper bound for the persistence exponent by considering the persistence probability on a suitable interval our main contribution is to prove this upper bound is the exact exponent and to extend to the whole real axis case

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

approximated penalized maximum likelihood for exploratory factor analysis an orthogonal case the problem of penalized maximum likelihood pml for an exploratory factor analysis efa model is studied in this paper an efa model is typically estimated using maximum likelihood and then the estimated loading matrix is rotated to obtain a sparse representation penalized maximum likelihood simultaneously fits the efa model and produces a sparse loading matrix to overcome some of the computational drawbacks of pml an approximation to pml is proposed in this paper it is further applied to an empirical dataset for illustration a simulation study shows that the approximation naturally produces a sparse loading matrix and more accurately estimates the factor loadings and the covariance matrix in the sense of having a lower mean squared error than factor rotations under various conditions

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

persistence exponents for fluctuating interfaces numerical and analytic results for the exponent ensuremath theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear langevin equations the models are parametrized by the dynamic roughness exponent ensuremath beta with 0l ensuremath beta l1 for ensuremath beta frac 1 2 the time evolution is markovian using simulations of solid on solid models of the discretized continuum equations as well as of the associated zero dimensional stationary gaussian process we address two problems the return of an initially flat interface and the return to an initial state with fully developed steady state roughness the two problems are shown to be governed by different exponents for the steady state case we point out the equivalence to fractional brownian motion which has a return exponent ensuremath theta s 1 ensuremath ensuremath beta the exponent ensuremath theta 0 for the flat initial condition appears to be nontrivial we prove that ensuremath theta 0 ensuremath rightarrow ensuremath infty for ensuremath beta ensuremath rightarrow 0 ensuremath theta 0 g ensuremath theta s for ensuremath beta l frac 1 2 and ensuremath theta 0 l ensuremath theta s for ensuremath beta g frac 1 2 and calculate ensuremath theta 0 s perturbatively to first order in an expansion around the markovian case ensuremath beta frac 1 2 using the exact result ensuremath theta s 1 ensuremath ensuremath beta accurate upper and lower bounds on ensuremath theta 0 can be derived which show in particular that ensuremath theta 0 g 1 ensuremath ensuremath beta 2 ensuremath beta for small ensuremath beta

deformation analysis of matrix models abstract in this paper we show that the non linear equations that play a role in tracy and widom s analysis of scaling limits for hermitian matrix models are special cases of the monodromy preserving deformation equations first discussed by sato miwa and jimbo in the early 1980 s

rotation to simple loadings using component loss functions the orthogonal case component loss functions clfs are used to generalize the quartimax criterion for orthogonal rotation in factor analysis these replace the fourth powers of the factor loadings by an arbitrary function of the second powers criteria of this form were introduced by a number of authors primarily katz and rohlf 1974 and rozeboom 1991 but there has been essentially no follow up to this work a method so simple natural and general deserves to be investigated more completely a number of theoretical results are derived including the fact that any method using a concave clf will recover perfect simple structure whenever it exists and there are methods that will recover thurstone simple structure whenever it exists specific clfs are identified and it is shown how to compare these using standardized plots numerical examples are used to illustrate and compare clf and other methods sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

integrable fredholm operators and dual isomonodromic deformations the fredholm determinants of a special class of integrable integral operators k supported on the union of m curve segments in the complex plane are shown to be the functions of an isomonodromic family of meromorphic covariant derivative operators having regular singular points at the 2m endpoints of the curve segments and a singular point of poincare index 1 at infinity the rank r of the corresponding vector bundle over the riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel the matrix riemann hilbert problem method is used to deduce an identification of the fredholm determinant as a function in the sense of segal wilson and sato i e in terms of abelian group actions on the determinant line bundle over a loop space grassmannian an associated dual isomonodromic family of covariant derivative operators having rank n 2m and r finite regular singular points located at the values of the exponents defining the kernel of k is derived the deformation equations for this family are shown to follow from an associated dual set of riemann hilbert data in which the roles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged the operators are analogously associated to an integral operator whose fredholm determinant is equal to that of k

newton algorithms for analytic rotation an implicit function approach in this paper implicit function based parameterizations for orthogonal and oblique rotation matrices are proposed the parameterizations are used to construct newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables the speed of the new algorithms is compared to that of existing algorithms and to that of newton algorithms based on alternative parameterizations several rotation criteria were examined and the algorithms were evaluated over a range of values for m initial guesses for newton algorithms were improved by subconvergence iterations of the gradient projection algorithm simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m among competing algorithms the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large in all other conditions however the implicit function algorithms were competitive with or faster than the fastest existing algorithms the new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

partition functions for matrix models and isomonodromic tau functions for one matrix models with polynomial potentials the explicit relationship between the partition function and the isomonodromic tau function for the 2 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived

bayesian exploratory factor analysis this paper develops and applies a bayesian approach to exploratory factor analysis that improves on ad hoc classical approaches our framework relies on dedicated factor models and simultaneously determines the number of factors the allocation of each measurement to a unique factor and the corresponding factor loadings classical identification criteria are applied and integrated into our bayesian procedure to generate models that are stable and clearly interpretable a monte carlo study confirms the validity of the approach the method is used to produce interpretable low dimensional aggregates from a high dimensional set of psychological measurements

derivative free gradient projection algorithms for rotation a simple modification substantially simplifies the use of the gradient projection gp rotation algorithms of jennrich 2001 2002 these algorithms require subroutines to compute the value and gradient of any specific rotation criterion of interest the gradient can be difficult to derive and program it is shown that using numerical gradients gives almost precisely the same results as using exact gradients the resulting algorithm is very easy to use because the only problem specific code required is that needed to define the rotation criterion the computing time is increased when using numerical gradients but it is still very modest for most purposes while used extensively elsewhere numerical derivatives seem to be underutilized in statistics

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

sparse versus simple structure loadings the component loadings are interpreted by considering their magnitudes which indicates how strongly each of the original variables relates to the corresponding principal component the usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value this in fact makes the component loadings sparse in an artificial and a subjective way we propose a new alternative approach which produces sparse loadings in an optimal way the introduced approach is illustrated on two well known data sets and compared to the existing rotation methods

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

extinction time of non markovian self similar processes persistence annihilation of jumps and the frechet distribution we start by providing an explicit characterization and analytical properties including the persistence phenomena of the distribution of the extinction time mathbb t of a class of non markovian self similar stochastic processes with two sided jumps that we introduce as a stochastic time change of markovian self similar processes for a suitably chosen time change we observe for classes with two sided jumps the following surprising facts on the one hand all the mathbb t s within a class have the same law which we identify in a simple form for all classes and reduces in the spectrally positive case to the frechet distribution on the other hand each of its distribution corresponds to the law of an extinction time of a single markov process without positive jumps leaving the interpretation that the time change has annihilated the effect of positive jumps the example of the non markovian processes associated to levy stable processes is detailed

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

large deviations for stationary gaussian processes in their previous work on large deviations the authors always assumed the base process to be markovian whereas here they consider the base process to be stationary gaussian similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process a rather explicit formula for the entropy involved is also obtained

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

no zero crossings for random polynomials and the heat equation consider random polynomial ni 0aixi of independent mean zero normal coefficients ai whose variance is a regularly varying function in i of order we derive general criteria for continuity of persistence exponents for centered gaussian processes and use these to show that such polynomial has no roots in 0 1 with probability n b o 1 and no roots in 1 with probability n b0 o 1 hence for n even it has no real roots with probability n 2b 2b0 o 1 here b 0 when 1 and otherwise b 0 is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary gaussian process of smooth sample path further making precise the solution d x t to the d dimensional heat equation initiated by a gaussian white noise d x 0 we confirm that the probability of d x t 0 for all t 1 t is t b o 1 for d 2 1

on semiclassical linear functionals integral representations abstract if is a functional of moments and satisfies the equation d o where x and x are arbitrary polynomials but with the only condition deg 1 then is said to be a semiclassical functional when is semiclassical and regular its corresponding orthogonal polynomial sequence ops pn x n 0 is called a semiclassical ops in this paper integral representations are given for semiclassical functionals regular or not in cases a deg deg and b deg deg but in the latter case when deg 0 the fundamental result is the following if n with n 0 are the moments of an a functional then n cm n this allows us to find integral representations for a functionals using standard techniques taken from pollaczek 1951