the gauss bonnet chern theorem a probabilistic perspective we prove that the euler form of a metric connection on a real oriented vector bundle e over a compact oriented manifold m can be identified as a current with the expectation of the random current defined by the zero locus of a certain random section of the bundle we also explain how to reconstruct probabilistically the metric and the connection on e from the statistics of random sections of e contents

large deviations of the maximum eigenvalue in wishart random matrices we analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the wishart laguerre ensemble of positive definite random matrices we show that the probability that all the eigenvalues of a n n wishart matrix w xtx where x is a rectangular m n matrix with independent gaussian entries are smaller than the mean value n c decreases for large n as where 1 2 corresponds respectively to real and complex wishart matrices c n m 1 and x c is a rate sometimes also called large deviation function that we compute explicitly the result for the anti wishart case m n simply follows by exchanging m and n we also analytically determine the average spectral density of an ensemble of wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier numerical simulations are in excellent agreement with the analytical predictions

metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections we prove the main conjecture from douglas shiffman and zelditch 2006 concerning the metric dependence and asymptotic minimization of the expected number ncrit n hof critical points of random holomorphic sections of the nth tensor power of a positive line bundle the first non topological term in the asymptotic expansion of ncrit n h is the calabi functional multiplied by the constant 2 m which depends only on the dimension of the manifold we prove that 2 m is strictly positive in all dimensions showing that the expansion is non topological for all m and that the calabi extremal metric when it exists asymptotically minimizes ncrit n h

numerical calabi yau metrics we develop numerical methods for approximating ricci flat metrics on calabi yau hypersurfaces in projective spaces our approach is based on finding balanced metrics and builds on recent theoretical work by donaldson we illustrate our methods in detail for a one parameter family of quintics we also suggest several ways to extend our results

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

a tower of coverings of quasi projective varieties abstract the main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non compact kahler manifold of finite volume with reasonable geometric assumptions to its universal covering examples to which our findings are applicable include moduli spaces of hyperbolic punctured riemann surfaces and hermitian locally symmetric spaces of finite volume

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

large deviations for gibbs measures with singular hamiltonians and emergence of kahler einstein metrics in the present paper and the companion paper berman kahler einstein metrics canonical random point processes and birational geometry arxiv 1307 3634 2015 a probabilistic statistical mechanical approach to the construction of canonical metrics on complex algebraic varieties x is introduced by sampling temperature deformed determinantal point processes the main new ingredient is a large deviation principle for gibbs measures with singular hamiltonians which is proved in the present paper as an application we show that the unique kahler einstein metric with negative ricci curvature on a canonically polarized algebraic manifold x emerges in the many particle limit of the canonical point processes on x in the companion paper berman in 2015 the extension to algebraic varieties x with positive kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for kahler einstein metrics with positive ricci curvature is developed

correlations between zeros and supersymmetry in our previous work bsz2 we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2 the purpose of this paper is to compute these universal limits in all dimensions and codimensions first we use a supersymmetry method to express the n point correlations as berezin integrals then we use the wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions

variance of the volume of random real algebraic submanifolds let mathcal x be a complex projective manifold of dimension n defined over the reals and let m denote its real locus we study the vanishing locus z s d in m of a random real holomorphic section s d of mathcal e otimes mathcal l d where mathcal l to mathcal x is an ample line bundle and mathcal e to mathcal x is a rank r hermitian bundle when r in 1 dots n 1 we obtain an asymptotic of order d r frac n 2 as d goes to infinity for the variance of the linear statistics associated to z s d including its volume given an open set u subset m we show that the probability that z s d does not intersect u is a o of d frac n 2 when d goes to infinity when n geq 3 we also prove almost sure convergence for the linear statistics associated to a random sequence of sections of increasing degree our framework contains the case of random real algebraic submanifolds of mathbb rp n obtained as the common zero set of r independent kostlan shub smale polynomials

recent progress on the quantum unique ergodicity conjecture we report on some recent striking advances on the quantum unique ergodicity or que conjecture concerning the distribution of large frequency eigenfunctions of the laplacian on a negatively curved manifold the account falls naturally into two categories the first concerns the general conjecture where the tools are more or less limited to micro local analysis and the dynamics of the geodesic flow the second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue the recent account by zelditch zel4 covers some of the same material and his and our discussion complement each other nicely as he goes into more detail with the first category and we with the second this note is not meant to be a survey of these topics and the discussion is not chronological our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context recent progress on que 2 1 the general que conjecture in figure 1 the domains e s and b are an ellipse a stadium and a barnett billiard table respectively superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions states modes of the laplacian that is they are solutions to 4 j j j 0 in 0 dirichlet boundary conditions j dxdy 1 0 here 4 divgrad 2 x2 2 y2 1 2 3 are the eigenvalues and the eigenfunctions are normalized to have unit l norm the sequences are of 12 consecutive modes around the 5600 eigenvalue they are ordered from left to right and then down and the grayscale represents the probability density with zero white and larger values darker

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

lowest landau level functional and bargmann spaces for bose einstein condensates abstract a fast rotating bose einstein condensate can be described by a complex valued wave function minimizing an energy restricted to the lowest landau level or fock bargmann space using some structures associated with this space we study the distribution of zeroes of the minimizer and prove in particular that the number of zeroes is infinite we relate their location to the combination of two problems a confining problem producing an inverted parabola profile and the abrikosov problem of minimizing an energy on a lattice using theta functions

asymptotic statistics of zeroes for the lame ensemble the lame polynomials naturally arise when separating variables in laplace s equation in elliptic coordinates the products of these polynomials form a class of spherical harmonics which are joint eigenfunctions of a quantum completely integrable qci system of commuting second order differential operators p 0 p 1 p n 1 acting on c n these operators naturally depend on parameters and thus constitute an ensemble in this paper we compute the limiting level spacings distributions for the zeroes of the lame polynomials in various thermodynamic asymptotic regimes we give results both in the mean and pointwise for an asymptotically full set of values of the parameters

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

critical points and supersymmetric vacua ii asymptotics and extremal metrics motivated by the vacuum selection problem of string m theory we study a new geometric invariant of a positive hermitian line bundle l h m over a compact kahler manifold the expected distribution k crit z of critical points dlog s z h 0 of a gaussian random holomorphic section s h 0 m l with respect to h it is a measure on m whose total mass is the average number n crit h of critical points of a random holomorphic section we are interested in the metric dependence of n crit h especially metrics h which minimize n crit h we concentrate on the asymptotic minimization problem for the sequence of tensor powers l n h n m of the line bundle and their critical point densities k critn z we prove that k critn z has a complete asymptotic expansion in n whose coefficients are curvature invariants of h the first two terms in the expansion of n crit hn are topological invariants of l m the third term is a topological invariant plus a constantm 2 depending only on the dimension m of m times the calabi functional r m 2 dv olh whereis the scalar curvature of the curvature form of h we give an integral formula form 2 and show by a computer assisted calculation thatm 2 0 for m 3 hence that n crit hn is asymptotically minimized by the calabi extremal metric when one exists we conjecture thatm 2 0 in all dimensions i e that the calabi extremal metric is always the asymptotic minimizer

density of complex critical points of a real random so m 1 polynomial we study the density of complex critical points of a real random so m 1 polynomial in m variables in a previous paper macdonald in j stat phys 136 5 807 2009 the author used the poincare lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials the su m 1 polynomials in this paper we use the kac rice formula to prove an analogous result the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial in one variable we give an exact formula and a scaling limit formula for the density of critical points of the real random so 2 polynomial as well as for the density of critical points of the corresponding complex random su 2 polynomial

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

near optimality of chebyshev interpolation for elementary function computations a common practice for computing an elementary transcendental function in an libm implementation nowadays has two phases reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval typically the interval is made tiny enough so that polynomials of very high degree aren t required for accurate approximations often approximating polynomials as such are taken to be the best polynomials or any others such as the chebyshev interpolating polynomials the best polynomial of degree n has the property that the biggest difference between it and the function is smallest among all possible polynomials of degrees no higher than n thus it is natural to choose the best polynomials over others in this paper it is proven that the best polynomial can only be more accurate by at most a fractional bit than the chebyshev interpolating polynomial of the same degree in computing elementary functions or in other words the chebyshev interpolating polynomials will do just as well as the best polynomials similar results were obtained in 1967 by powell who however did not target elementary function computations in particular and placed no assumption on the function and remarkably whose results imply accuracy differences of no more than 2 to 3 bits

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

poincare lelong approach to universality and scaling of correlations between zeros this note is concerned with the scaling limit as n of n point correlations between zeros of random holomorphic polynomials of degree n in m variables more generally we study correlations between zeros of holomorphic sections of powers ln of any positive holomorphic line bundle l over a compact kahler manifold distances are rescaled so that the average density of zeros is independent of n our main result is that the scaling limits of the correlation functions and more generally of the correlation forms are universal i e independent of the bundle l manifold m or point on m

equidistribution and convergence speed for zeros of holomorphic sections of singular hermitian line bundles abstract we establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular hermitian line bundle with an estimate of the convergence speed

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

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

bergman kernels and equilibrium measures for line bundles over projective manifolds let l be a holomorphic line bundle over a compact complex projective hermitian manifold x any fixed smooth hermitian metric h on l induces a hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of l in this paper various convergence results are obtained for the corresponding bergman kernels i e orthogonal projection kernels the convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h e associated to h as well as in terms of the monge ampere measure of h on a certain support set it is also shown that the equilibrium metric h e is in the class c 1 1 on the complement of the augmented base locus of l for l ample these results give generalizations of well known results concerning the case when the curvature of h is globally positive then h e h in general the results can be seen as local metrized versions of fujita s approximation theorem for the volume of l

random polynomials with prescribed newton polytope the newton polytope pf of a polynomial f is well known to have a strong impact on its zeros as in the kouchnirenko bernstein theorem on the number of simultaneous zeros of m polynomials with given newton polytopes in this article we show that pf also has a strong impact on the distribution of zeros of one or several polynomials we equip the space of holomorphic polynomials of degreen in m complex variables with its usual su m 1 invariant gaussian measure and then consider the conditional measures np induced on the subspace of polynomials whose newton polytope pfnp when p the unit simplex then it is obvious and well known that the expected distribution of zeros zf1 fk regarded as a current of k polynomials f1 fk of degree n is uniform relative to the fubini study form our main results concern the conditional expectation e np zf1 fk of zeros of k polynomials with newton polytope npnp where p deg p these results are asymptotic as the scaling factor n 1 we show that e np zf1 fk is asymptotically uniform on the inverse image ap of the open scaled polytope 1 p via the moment map cp m for projective space however the zeros have an exotic distribution outside of ap and when k m the case of the kouchnirenko bernstein theorem the percentage of zeros outside ap approaches 0 as n 1

large deviations for zeros of p ϕ 2 random polynomials we extend the results of zeitouni and zelditch in int math res not 2010 20 3939 3992 2010 on ldps large deviations principles for the empirical measures z s frac 1 n sum zeta s zeta 0 delta zeta quad n zeta s zeta 0 of zeros of gaussian random polynomials s in one variable to p 2 random polynomials the speed and rate function are the same as in the associated gaussian case it follows that the expected distribution of zeros in the p 2 ensembles tends to the same equilibrium measure as in the gaussian case

a unifying integral iss framework for stability of nonlinear cascades we analyze nonlinear cascades in which the driven subsystem is integral input to state stable iss and we characterize the admissible integral iss gains for stability this characterization makes use of the convergence speed of the driving subsystem and allows a larger class of gain functions when the convergence is faster we show that our integral iss gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem the result is used to develop a new observer based backstepping design in which the growth of the nonlinear damping terms is reduced

formally verified argument reduction with a fused multiply add the cody and waite argument reduction technique works perfectly for reasonably large arguments but as the input grows there are no bits left to approximate the constant with enough accuracy under mild assumptions we show that the result computed with a fused multiply add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision we also present an algorithm for a fully accurate second reduction step to reach full double accuracy all the significand bits of two numbers are accurate even in the worst cases of argument reduction our work recalls the common algorithms and presents proofs of correctness all the proofs are formally verified using the coq automatic proof checker

equidistribution of zeros of random holomorphic sections for moderate measures we establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle the equidistribution is associated with a family of singular moderate measures we also give a convergence speed for the equidistribution

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

critical points of random branched coverings of the riemann sphere given a closed riemann surface sigma equipped with a volume form omega we construct a natural probability measure on the space mathcal m d sigma of degree d branched coverings from sigma to the riemann sphere mathbb c mathbb p 1 we prove a large deviations principle for the number of critical points in a given open set u subset sigma that is given any sequence epsilon d of positive numbers the probability that the number of critical points of a branched covering deviates from 2d cdot text vol u more than epsilon d cdot d is smaller than exp c u epsilon 3 d d for some positive constant c u in particular the probability that a covering does not have any critical point in a given open set goes to zero exponential fast with the degree

parametric statistics of zeros of husimi representations of quantum chaotic eigenstates and random polynomials local parametric statistics of zeros of husimi representations of quantum eigenstates are introduced it is conjectured that for a classically fully chaotic system one should use the model of parametric statistics of complex roots of gaussian random polynomials which is exactly solvable as demonstrated below for example the velocities derivatives of zeros of husimi function with respect to an external parameter are predicted to obey a universal non maxwellian distribution where is the mean square velocity the conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom dynamical formulation of the zero flow in terms of an integrable many body dynamical system is given as well

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

the distribution of the zeros of random trigonometric polynomials we study the asymptotic distribution of the number z n of zeros of random trigonometric polynomials of degree n as n rightarrow infty it is known that as n grows to infinity the expected number of the zeros is asymptotic to frac 2 sqrt 3 cdot n the asymptotic form of the variance was predicted by bogomolny bohigas and leboeuf to be cn for some c 0 we prove that frac z n bbb e z n sqrt cn converges to the standard gaussian in addition we find that the analogous result is applicable for the number of zeros in short intervals

random riesz energies on compact kahler manifolds the expected riesz energies e n h es of the zero sets of systems of independent gaussian random polynomials are determined asymptotically as the degree n in all dimensions and codimensions the asymptotics are proved for sections of any positive line bundle over any compact kahler manifold

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

equivariant local scaling asymptotics for smoothed toplitz spectral projectors abstract let x be the unit circle bundle of a positive line bundle on a hodge manifold we study the local scaling asymptotics of the smoothed spectral projectors associated with a first order elliptic toplitz operator t on x possibly in the presence of hamiltonian symmetries the resulting expansion is then used to give a local derivation of an equivariant weyl law it is not required that t be invariant under the structure circle action that is t needn t be a berezin toplitz operator

an asymptotic expansion for the expected number of real zeros of real random polynomials spanned by opuc abstract let i i 0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive borel measure that is symmetric with respect to conjugation we study asymptotic behavior of the expected number of real zeros say e n of random polynomials p n z i 0 n i i z where 0 n are i i d standard gaussian random variables when is the acrlength measure such polynomials are called kac polynomials and it was shown by wilkins that e n d admits an asymptotic expansion of the form e n d 2 log n 1 p 0 a p n 1 p kac himself obtained the leading term of this expansion in this work we generalize the result of wilkins to the case where is absolutely continuous with respect to arclength measure and its radon nikodym derivative extends to a holomorphic non vanishing function in some neighborhood of the unit circle in this case e n admits an analogous expansion with the coefficients a p depending on the measure for p 1 the leading order term and a 0 remain the same

critical points and supersymmetric vacua i supersymmetric vacua universes of string m theory may be identified with certain critical points of a holomorphic section the superpotential of a hermitian holomorphic line bundle over a complex manifold an important physical problem is to determine how many vacua there are and how they are distributed as the superpotential varies over physically relevant ensembles in several papers over the last few years m r douglas and co workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor do ad dd the present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem it sets down basic results on the statistics of critical points s 0 of random holomorphic sections of hermitian holomorphic line bundles with respect to a metric connection when the sections are endowed with a gaussian measure the principal results give formulas for the expected density and number of critical points of fixed morse index of gaussian random sections relative to they are particularly concrete for riemann surfaces in our subsequent work the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points

two point eigenvalue correlation for a class of non selfadjoint operators under random perturbations we consider a non selfadjoint h differential model operator ph in the semiclassical limit h rightarrow 0 subject to random perturbations with a small coupling constant assume that rm e frac 1 ch text 0 suitably large let be the closure of the range of the principal symbol

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

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

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

observer design for systems with multivariable monotone nonlinearities globally convergent observers are designed for a class of systems with multivariable nonlinearities the approach is to represent the observer error system as the feedback interconnection of a linear system and a state dependent multivariable nonlinearity we first extend an earlier design automatica 37 12 2001 1923 to multivariable nonlinearities satisfying an analog of the scalar nondecreasing property next we exploit the structure of the nonlinearity to relax the positive real restriction on the linear part of the observer error system this relaxed design renders the feasibility conditions less restrictive and widens the applicability of the observer as illustrated with examples finally output nonlinearities are studied and the design is extended to be adaptive in the presence of unknown parameters

critical points and supersymmetric vacua iii string m models a fundamental problem in contemporary string m theory is to count the number of inequivalent vacua satisfying constraints in a string theory model this article contains the first rigorous results on the number and distribution of supersymmetric vacua of type iib string theories compactified on a calabi yau 3 fold x with flux in particular complete proofs of the counting formulas in ashok douglas ad and denef douglas dd1 are given together with van der corput style remainder estimates

asymptotics and dimensional dependence of the number of critical points of random holomorphic sections we prove two conjectures from dsz2 dsz3 concerning the expected number of critical points of random holomorphic sections of a positive line bundle we show that on average the critical points of minimal morse index are the most plentiful for holomorphic sections of mathcal o n to mathbb cp m and in an asymptotic sense for those of line bundles over general kahler manifolds we calculate the expected number of these critical points for the respective cases and use these to obtain growth rates and asymptotic bounds for the total expected number of critical points in these cases this line of research was motivated by landscape problems in string theory and spin glasses

universality results for zeros of random holomorphic sections in this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular hermitian holomorphic line bundles on a compact kahler complex space x namely under mild moment assumptions we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections in the case when x is a compact kahler manifold we also prove an off diagonal exponential decay estimate for the bergman kernels of a sequence of positive line bundles on x

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

gauss bonnet chern theorem on moduli space in this paper we prove a gauss bonnet chern type theorem in full generality for the chern weil forms of hodge bundles that is the chern weil forms compute the corresponding chern classes this settles a long standing problem second we apply the result to calabi yau moduli and proved the corresponding gauss bonnet chern type theorem in the setting of weil petersson geometry as an application of our results in string theory we prove that the number of flux vacua of type ii string compactified on a calabi yau manifold is finite and their number is bounded by an intrinsic geometric quantity

scaled correlations of critical points of random sections on riemann surfaces in this paper we prove that as n goes to infinity the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the n th power of a positive line bundle over a compact riemann surface tends to 2 3 2 for small sqrt n z 1 nobreak z 2 the scaling limit is directly calculated using a general form of the kac rice formula and formulas and theorems of pavel bleher bernard shiffman and steve zelditch

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

canonical singular hermitian metrics on relative canonical bundles we introduce a new class of canonical analytic zariski decompositions azd s in short called the supercanonical azd s on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes we study the variation of the supercanonical azd hat h rm can under projective deformations and give a new proof of the invariance of plurigenera

global output tracking control of a class of euler lagrange systems with monotonic non linearities in the velocities in this paper we address the problem of output feedback tracking control of a class of euler lagrange systems subject to non linear dissipative loads by imposing a monotone damping condition on the non linearities of the unmeasured states the common restriction that the non linearities be globally lipschitz is removed the proposed observer controller scheme renders the origin of the error dynamics uniformly globally asymptotically stable in the general case under certain additional assumptions the result continue to hold for a simplified control law that is less sensitive to noise and unmodelled phenomena

a new range reduction algorithm range reduction is a key point for getting accurate elementary function routines we introduce a new algorithm that is fast for input arguments belonging to the most common domains yet accurate over the full double precision range

density of complex zeros of a system of real random polynomials we study the density of complex zeros of a system of real random so m 1 polynomials in m variables we show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case we also show that the behavior the scaled density of complex zeros near m of the system of real random polynomials is different in the m 2 case than in the m 1 case the density approaches infinity instead of tending linearly to zero

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n

local trace formulae and scaling asymptotics for general quantized hamiltonian flows a class of trace formulae in the toeplitz quantization of hamiltonian flows on compact symplectic manifolds is given a local intepretation in terms of certain scaling asymptotics near a symplectic fixed locus these are reminiscent of the near diagonal asymptotics of equivariant szego kernels in spectral theory the distributional trace of a positive elliptic operator encapsulates information on the asymptotic distribution of its eigenvalues a trace formula relates the singularities of the trace to poincare data of an appropriate hamiltonian flow this leads to the asymptotics of certain smoothing kernels related to the fourier transform of the trace of a wave operator in toeplitz quantization we consider analogues of these with the wave operator replaced by a quantized hamiltonian flow possibly composed with a family of zeroth order toeplitz operators we study the concentration of these smoothing kernels near the fixed loci of the linearized dynamics and specifically how the poincare map controls their scaling asymptotics this generalizes previous results on holomorphic flows

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions