Matrix Measures, Random Moments, and Gaussian Ensembles

We consider the moment space M_n\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S_1,n, ... , S_n,n)^* ~ U (M_n)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S _1,n ,…,S _k,n )^∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space M_n\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.     

From Gumbel to Tracy-Widom

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( ? exp( ? x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp( ? x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $M=M_0+\sqrt{2S} V$ , with M ₀ diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

Operators associated with soft and hard spectral edges from unitary ensembles

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy-Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy-Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann-Weyl anti-commutation relations and leave invariant the subspaces of L² that are the ranges of projections given by the Tracy-Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.     

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Hausmann and Knutson, using the Hopf map on quaternions from the complex Stiefel manifold of 2‐frames in n‐space to the space of closed n‐gons in 3‐space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons that comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges. © 2014 Wiley Periodicals, Inc.     

Homogeneous random fields and statistical mechanics

We illustrate the connection between homogeneous perturbations of homogeneous Gaussian random fields over Rⁿ or Zⁿ, with values in R^m, and classical as well as quantum statistical mechanics. In particular we construct homogeneous non-Gaussian random fields as weak limits of perturbed Gaussian random fields and study the infinite volume limit of correlation functions for a classical continuous gas of particles with inner degrees of freedom. We also exhibit the relation between quantum statistical mechanics of lattice systems (anharmonic crystals) at temperature β^?1 and homogeneous random fields over Zⁿ × S_β, where S_β is the circle of length β, which then provides a connection also with classical statistical mechanics. We obtain the infinite volume limit of real and imaginary times Green's functions and establish its properties. We also give similar results for the Gibbs state of the correspondent classical lattice systems and show that it is the limit as h → 0 of the quantum statistical Gibbs state.     

Eigenvector distribution of Wigner matrices

We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν _ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν _ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.     

Une méthode semi-paramétrique pour tester un modèle de régression

The purpose is to test the hypothesis H₀ that a regression model is parametric and belongs to a given family versus the alternative H₁ approaches the hypothesis from a specific direction at the rate n^?1/2. For that, we consider an empirical process such that under H₀ this process depends of a parameter θ₀. First, we start by estimating the parameter and we prove that the empirical process converges in distribution to a certain Gaussian process when the parameter is replaced by its estimator $\text{θ}{}_{\mathrm{n}}\text{.}$ However it is important to check the impact of an alternative approaching H₀ from a specific direction ( at the rate n^1/2). For that, we need tests which are consistent on the whole of H₁. Our idea is to use a marked empirical process based on residuals which converges in distribution to a Gaussian process. To cite this article: M. Harel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

Global properties of tensor rank

The dependence of tensor rank on the underlying ring of scalars is considered. It is shown that the integers are, in a certain sense, the worst scalars. A ring of scalars can be improved by adjoining algebraic elements but not by adjoining indeterminates. The real closed fields are the best scalars among ordered rings, and the algebraically closed fields are best among all rings. Let B(R^m×n×p) be the maximum tensor rank of any m×n×p array of elements from the ring R. A generalization of Gaussian elimination shows that B(R^n×n×n)?$\text{3}\text{4}$n² for most useful rings R. For every R, B(R^m×n×p)?mnp/(m+n+p), and slightly stronger lower bounds are proven for R a field.     

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Given an ensemble of N×N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N→∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f _m show a visually stunning convergence to the semi-circle as m→∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f _m is the product of a Gaussian and a certain even polynomial of degree 2m?2; the formula is the same as that for the m×m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.     

Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ?: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² ? r ⁿ ? 3r² ? 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.     

On fractional Hadamard powers of positive definite matrices

Let A = (a_ij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that A^α ≡ (a_ij^α) is positive definite for all real α ? n ? 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix.     

Equidistribution of signs for modular eigenforms of half integral weight

Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp ²)} _p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn ²)} _n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form [0, n] × [?h _n , h _n ] ^d-1 obey Gaussian central limit theorems as long as h _n grows slower than n ^1/(d+1). It is an open question as to what is the fastest that h _n can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of (1, 0, . . . , 0), we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.     

Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C² as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C² in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).     

Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.