On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

Abstract

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τ -functions.     

Fredholm Determinants and the Statistics of Charge Transport

Using operator algebraic methods we show that the moment generating function of charge transport in a system with infinitely many non-interacting Fermions is given by a determinant of a certain operator in the one-particle Hilbert space. The formula is equivalent to a formula of Levitov and Lesovik in the finite dimensional case and may be viewed as its regularized form in general. Our result embodies two tenets often realized in mesoscopic physics, namely, that the transport properties are essentially independent of the length of the leads and of the depth of the Fermi sea.     

Singular Harmonic Maps and Applications to General Relativity

The study of axially symmetric stationary multi-black-hole configurations and the force between co-axially rotating black holes involves, as a first step, an analysis on the “boundary regularity” of the so-called reduced singular harmonic maps. We carry out this analysis by considering those harmonic maps as solutions to some homogeneous divergence systems of partial differential equations with singular coefficients. Our results extend previous works by Weinstein (Comm Pure Appl Math 43:903–948, 1990; Comm Pure Appl Math 45:1183–1203, 1992) and by Li and Tian (Manu Math 73(1):83–89, 1991; Commun Math Phys 149:1–30, 1992; Differential geometry: PDE on manifolds, vol 54, pp. 317–326, 1993). This paper is based on the Ph.D. thesis of the author (Singular harmonic maps into hyperbolic spaces and applications to general relativity, PhD thesis, The State University of New Jersey, Rutgers, 2009).     

Multichannel Coupled Nonlocal Interactions and Generalized Levinson Theorem

The T matrix is determined by the generalized Lipprnann-Schwinger equations with the multichannel coupled nonlocal separable potential of arbitrary rank N, and for the potentials, a generalized Levinson theorem is proved with a new method.     

Solution to Operator Fredholm Equation in Bipartite Entangled State Representation and Its Applications

Based on the bipartite entangled state representation and using the technique of integration within an ordered product （IWOP） of operators we construct the corresponding operator Fredholm equations and then derive their solutions. As its application we deduce some new bosonic operator identities and new relations about the two-variable Hermite polynomials.     

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.     

Nodal deficiency,spectral flow,and the Dirichlet-to-Neumann map

Letters in Mathematical Physics - It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the...     

Normally Ordered Operator Fredholm Equation and Its Applications in Quantum Mechanics

By virtue of the technique of integration within an ordered product of operators we construct the normally ordered operator fiedholm equation. We use it to derive some new operator formulas. For Weyl correspondence, operator fiedholm equation can also be constructed. Some applications of the operator Fkedholm equation are given.     

Digital Bispectral Analysis and Its Applications to Nonlinear Wave Interactions

The bispectrum, which is an ensemble average of a product of three spectral components, is shown to be a very useful diagnostic tool in experimental studies of nonlinear wave interactions in random media. In particular, it is shown that the bicoherence spectrum may be used to discriminate between nonlinearly coupled waves and spontaneously excited waves and to measure the fraction of wave power due to the quadratic wave coupling in a self-excited fluctuation spectrum. Practical aspects of digital bispectral analysis techniques, such as estimation and statistical variability of the estimator, are also discussed. Finally, applications of bispectral analysis techniques in the analysis and interpretation of plasma fluctuation data are described.     

Thermal Nonlocal Advantage of Quantum Coherence in the Two-Site,Triangular, and Tetrahedral Lattices with Heisenberg Interactions

International Journal of Theoretical Physics - Quantum correlations are physical resources for quantum information processing. The nonlocal advantage of quantum coherence (NAQC) is a kind of...     

Nonlocal density functional theory of solids: Applications of the weighted density approximation to silicon

Using the density functional formalism together with the weighted density approximation (WDA), we have carried out selfconsistent bandstructure studies of bulk Si in order to identify the influences of different WDA potentials and different pair correlation functions on the electronic spectrum. Improvements of band energies due to consideration of nonlocalities in exchange and correlation energy depend very sensitively on the particular correlation function used. For the prototype semiconductor Si, only one of the six WDA versions tested seems useful.     

On Quantum Correlations and Positive Maps

We present a discussion on local quantum correlations and their relations with entanglement. We prove that a vanishing coefficient of quantum correlations implies separability. The new results on locally decomposable maps which we obtain in the course of the proof also seem to be of independent interest.     

Liquid-Vapor Interfaces and Surface Tension in a Mesoscopic Model of Fluid with Nonlocal Interactions

We analyze the problem of phase coexistence, surface tension and the interface patterns between liquid and vapour for the nonlocal free energy functional derived by Lebowitz, Mazel, and Presutti from a system of particles interacting through Kac potentials in the continuum. We study the sharp interface limit in d dimensions and characterize the shape of the interface profiles in the temperature region where a monotonicity property is valid. We further extend our analysis beyond this domain by performing numerical simulations.     

简介生物学领域中的拉曼散射研究

本文叙述了用拉曼光谱进行生物分子和体系研究的优点，并从生物体系的结构、功能和动力学三方面介绍了拉曼光谱研究工作所取得的一些进展。     

Extended Rearrangement Inequalities and Applications to Some Quantitative Stability Results

In this paper, we prove a new functional inequality of Hardy–Littlewood type for generalized rearrangements of functions. We then show how this inequality provides quantitative stability results of steady states to evolution systems that essentially preserve the rearrangements and some suitable energy functional, under minimal regularity assumptions on the perturbations. In particular, this inequality yields a quantitative stability result of a large class of steady state solutions to the Vlasov–Poisson systems, and more precisely we derive a quantitative control of the L¹ norm of the perturbation by the relative Hamiltonian (the energy functional) and rearrangements. A general non linear stability result has been obtained by Lemou et al. (Invent Math 187:145–194, 2012) in the gravitational context, however the proof relied in a crucial way on compactness arguments which by construction provides no quantitative control of the perturbation. Our functional inequality is also applied to the context of 2D-Euler systems and also provides quantitative stability results of a large class of steady-states to this system in a natural energy space.     

On Factor Maps that Send Markov Measures to Gibbs Measures

Let X and Y be mixing shifts of finite type. Let π be a factor map from X to Y that is fiber-mixing, i.e., given \(x,\bar{x}\in X\) with \(\pi(x)=\pi(\bar{x})=y\in Y\), there is z∈π ^?1(y) that is left asymptotic to x and right asymptotic to \(\bar{x}\). We show that any Markov measure on X projects to a Gibbs measure on Y under π (for a Hölder continuous potential). In other words, all hidden Markov chains (i.e. sofic measures) realized by π are Gibbs measures. In 2003, Chazottes and Ugalde gave a sufficient condition for a sofic measure to be a Gibbs measure. Our sufficient condition generalizes their condition and is invariant under conjugacy and time reversal. We provide examples demonstrating our result.     

Fredholm determinants,differential equations and matrix models

Orthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form ((x)(y)–(x)(y))/x–y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals . The emphasis is on the determinants thought of as functions of the end-pointsa _k.We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as and satisfy a certain type of differentiation formula. The (, ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finiteN Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system.An analysis of these equations will lead to explicit representations in terms of Painlevé transcendents for the distribution functions of the largest and smallest eigenvalues in the finiteN Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables.There is also an exponential variant of the kernel in which the denominator is replaced bye ^bx–e^by, whereb is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. Ifb=i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble ofN×N unitary matrices, and then an ODE ifJ is an arc of the circle.     

On the Bilinearisation Approach to the Soliton Solutions of Cylindrical and Nonlocal K-P Equation

We have obtained the N-soliton solution for the two nonlinear partial differential equations — the cylindrical K-P equation and non-local K-P equation. The phenomenon of soliton resonance is demonstrated in the case of cylindrical K-P equation.