On the universality of the level spacing distribution for some ensembles of random matrices

We prove that the level spacing distribution at the middle of the spectrum of some one-parameter family of random matrix ensembles has the universal form coinciding with that previously known for several special ensembles. We also discuss some related topics of the random matrix theory.     

Continuum models of two-dimensional random droplets

We study the statistical mechanical properties of a two-dimensional assembly of free particles coupled to a mechanical reservoir. The particles-reservoir interaction is modelised by an Hamiltonian depending on the convex hull of the particles only. We concentrate on models whose energy is the sum of an area-term, a perimeter term and possibly a term preventing the particles occupying the interior of the convex hull. The range of coupling constants insuring a thermodynamic behaviour, as well as the associated free energy per particle are exactly determined.Work partially supported by the Swiss National Foundation for Scientific Research.     

Higher trace and Berezinian of matrices over a Clifford algebra

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded commutative associative algebra A$A$. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded matrices of degree 0$0$ is polynomial in its entries. In the case of the algebra A=H$A=\mathbb{H}$ of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded version of Liouville’s formula.     

An extension of the Ccayley-Hamilton theorem to the case of supermatrices

Starting from the expression for the superdeterminant of (xI - M), whereM is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the superdeterminant, we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix.     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Geometry of quadrilateral nets: Second Hamiltonian form

Discrete Darboux–Manakov–Zakharov systems possess two distinct Hamiltonian forms (by this term we mean that equations of motion are discrete time extensions of Hamiltonian equations of motion). In the framework of discrete-differential geometry one Hamiltonian form appears in a geometry of a circular net. In this paper a geometry of the second form is identified.     

Distinguished Hamiltonian theorem for homogeneous symplectic manifolds

A diffeomorphism of a finite-dimensional flat symplectic manifold which is canonoid with respect to all linear and quadratic Hamiltonians, preserves the symplectic structure up to a factor: so runs the quadratic Hamiltonian theorem. Here we show that the same conclusion holds for much smaller sufficiency subsets of quadratic Hamiltonians, and the theorem may thus be extended to homogeneous infinite-dimensional symplectic manifolds. In this way, we identify the distinguished Hamiltonians for the Kähler manifold of equivalent quantizations of a Hilbertizable symplectic space.     

P-Invertibility of matrices and operators

We prove a multiplicity theorem which replaces a variety of rules used in the theory of the intermediate problem of the first type for eigenvalues of semi-bounded self-adjoint operators on a complex Hilbert space.     

Asymptotic properties of the massive scalar field in the external Schwarzschild spacetime

The asymptotic properties of the solution to the Klein–Gordon equation will be studied in the Schwarzschild spacetime background. The results are based on the global Sobolev-type inequalities and the generalized energy estimates.     

On the failure of subadditivity of the Wigner–Yanase entropy

It was recently shown by Hansen that the Wigner–Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting applications in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample. Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. This paper may be reproduced, in its entirety, for non-commerical purposes.     

Asymptotic properties of a simple random motion

A random walker in ^N is considered. At each step the walker picks a point in ^N from a fixed finite set of destination points. Having chosen the point, the walker moves a fractionr (r<1) of the distance toward the point along a straight line. Assuming that the successive destination points are chosen independently, it is shown that the asymptotic distribution of the walker's position has the same mean as the destination point distribution. An estimate is obtained for the fraction of time the walker stays within a ball centered at the mean value for almost every destination sequence. Examples show that the asymptotic distribution could have intricate structure.     

The central extension of Kac-Moody-Malcev algebras

We introduce a class of infinite-dimensional Kac-Moody-Malcev algebras. These algebras are the generalization of Lie algebras of the Kac-Moody type to Malcev algebras. We demonstrate that the central extensions of the Kac-Moody-Malcev algebras are given by the same cocycles as in the case of Lie algebras. Analogues of Kac-Moody-Malcev algebras may be also introduced in the case of an arbitrary Riemann surface.     

Infinite products of large random matrices and matrix-valued diffusion

We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large Hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of Hermitian and complex matrices, we observe the emergence of a “topological phase transition”, when a hole develops in the eigenvalue spectrum, after some critical diffusion time τ_crit is reached. In the case of a particular product of two Hermitian ensembles, we observe also an unusual localization–delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulas obtained in this work by numerical simulation.     

On pseudo-continuities in the eigenvalue continuum of one-speed neutron transport theory

We report on a new closed-form approximant to the singular eigenfunction transient solution of the one-speed, one-dimensional neutron transport equation with an anisotropic scattering kernel. It is proved that the associated eigenvalue continuum must be pointwise perforated.     

Symplectic structures on moduli spaces of sheaves via the Atiyah class

It is proven that the composition of the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves. Two examples are given when this 2-form is symplectic. Both of them are moduli spaces of torsion sheaves on the cubic 4-fold Y$Y$. The first example is the Fano scheme of lines in Y$Y$. Beauville and Donagi showed that it is symplectic but did not construct an explicit symplectic form on it. We prove that our construction provides a symplectic form. The other example is the moduli space of torsion sheaves which are supported on the hyperplane sections H∩Y$H\cap Y$ of Y$Y$ and are cokernels of the Pfaffian representations of H∩Y$H\cap Y$.     

Asymptotic properties of multistate random walks. I. Theory

A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) formultistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of a periodically repeated unit cell which contains a finite number of internal states (sites). The results are identical to those for perfect lattices except for a renormalization of coefficients. For walks without drift, it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk. The diffusion coefficients can be obtained by a simple matrix algorithm presented here. Both discrete and continuous time random walks are considered. The results are not restricted to nearest-neighbor random walks but apply as long as the single-step probability distributions associated with each of the internal states have finite means and variances.     

On the limit of extreme eigenvalues of large dimensional random quaternion matrices

Since E.P. Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case.     

The lie bialgebroid of a Poisson-Nijenhuis manifold

We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures.     

On exterior boundary-value problems of the Helmholtz equation not deduced from waves

A boundary integral equation is applied to describe a special kind of exterior Helmholtz boundary-value problem that is not deduced from waves. Then the asymptotic property of O(r ^–2) decay at infinity and the uniqueness of the solution as well as its finite energy property are discussed.