量子不可积性和随机矩阵理论

本文概述了量子不可积性的拓扑观.按拓扑观点探讨了量子体系完全可积的条件,通向不可积的机制,以及在量子不可积条件下的能谱性质,从动力学角度阐明了随机矩阵理论的基础,还简要讨论了这些理论在核结构问题中的应用. Quantum nonintegrability is studied in a topological view.The condition for complete integrability of quantum systems and the mechanism leading to global nonintegrability are investigated.The basis of random matrix theory for describing energy spectra of nonintegrable quantum systems is clarified.Applications to nuclear structure are briefly discussed.     

Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in \({(\mathbb {C}^d)^{\otimes k}}\), where k and p/d ^k are fixed while d → ∞. When k = 1, the Mar?enko-Pastur law determines (up to small corrections) not only the largest eigenvalue (\({(1+\sqrt{p/d^k})^2}\)) but the smallest eigenvalue \({(\min(0,1-\sqrt{p/d^k})^2)}\) and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately \({(1+\sqrt{p/d^k})^2}\) and the spectral density approaches that of the Mar?enko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter.Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.     

Random Matrix Theory and the Anderson Model

This paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal-insulator Anderson transition for the Anderson model in dimension d≥3. The possible criterions used to define such a transition are presented. It is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model. Various simplified versions for which rigorous results have been obtained in the past are discussed. It includes a free probability approach, the Wegner n-orbital model and a class of models proposed by Disertori, Pinson, and Spencer, Comm. Math. Phys. 232:83–124 (2002). At last a recent work by Magnen, Rivasseau, and the author, Markov Process and Related Fields 9:261–278 (2003) is summarized: it gives a toy modeldescribing the lowest order approximation of Anderson model and it is proved that, for d=2, its density of states is given by the semicircle distribution. A short discussion of its extension to d≥3 follows.     

Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.Acknowledgement We gratefully acknowledge stimulating discussions with Estelle Basor, Peter Forrester and Noah Linden. We are also grateful for the kind hospitality of the Isaac Newton Institute for the Mathematical Sciences, Cambridge, while this research was completed. Francesco Mezzadri was supported by a Royal Society Dorothy Hodgkin Research Fellowship.     

A Random Matrix Model From Two Dimensional Yang-Mills Theory

We use fat graphs to give a unified treatment of various asymptotic freeness results. In particular, a random matrix model from two dimensional Yang-Mills Theory on the plane is presented. Received: 10 June 1996 / Accepted: 6 April 1997     

Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy

Using random matrix techniques and the theory of Matrix Product States we show that reduced density matrices of quantum spin chains have generically maximum entropy.     

Random Matrix Theory and the Boson Peak in Two-Dimensional Systems

Physics of the Solid State - The random matrix theory is used to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown...     

Superbosonization Formula and its Application to Random Matrix Theory

Starting from Gaussian random matrix models we derive a new supermatrix field theory model. In contrast to the conventional non-linear sigma models, the new model is applicable for any range of correlations of the elements of the random matrices. We clarify the domain of integration for the supermatrices, and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices. It is also shown how one can reduce the supermatrix model to the conventional sigma model.     

Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function F _nm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and \frac1nmF_nm{\frac{1}{nm}F_{nm}} are the Grunsky coefficients of the Faber polynomials.     

Label-Free Microscopic Imaging Based on the Random Matrix Theory in Wavefront Shaping

Wavefront shaping technology has mainly been applied to microscopic fluorescence imaging through turbid media,with the advantages of high resolution and imaging depth beyond the ballistic regime. However, fluorescence needs to be introduced extrinsically and the field of view is limited by memory effects. Here we propose a new method for microscopic imaging light transmission through turbid media, which has the advantages of label-free and discretional field of view size, based on transmission-matrix-based wavefront shaping and the random matrix theory. We also verify that a target of absorber behind the strong scattering media can be imaged with high resolution in the experiment. Our method opens a new avenue for the research and application of wavefront shaping.     

Moment Matrices and Multi-Component KP,with Applications to Random Matrix Theory

Questions on random matrices and non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The main result of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of “time” deformations for each weight, the multi-component KP-hierarchy: these determinants are thus “tau-functions” for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. The main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials: it concerns an identity between the orthogonality of polynomials on the real line, the bilinear identity in KP theory and a generating functional for the full KP theory. An additional fact not discussed in this paper is that these τ-functions satisfy Virasoro constraints with respect to these time parameters. As one of the many examples worked out in this paper, we consider N non-intersecting Brownian motions in leaving from the origin, with n _i particles forced to reach p distinct target points b _i at time t  =  1; of course, . We give a PDE, in terms of the boundary points of the interval E, for the probability that the Brownian particles all pass through an interval E at time 0  <  t  <  1. It is given by the determinant of a (p + 1)  ×  (p + 1) matrix, which is nearly a wronskian. This theory is also applied to biorthogonal polynomials and orthogonal polynomials on the circle. The support of a National Science Foundation grant # DMS-07-04271 is gratefully acknowledged. The support of a National Science Foundation grant # DMS-07-04271, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), a FNRS grant and a “Interuniversity Attraction Pole” grant is gratefully acknowledged. The support of a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA) and a ANR grant (GIMP) is gratefully acknowledged.     

Quasiclassical Theory of Phase Relaxation by Gauge Field Fluctuations

The quasiclassical theory in terms of Feynman path integrals is used to calculate the decay of the Cooperon amplitude caused by transverse gauge field fluctuations in a disordered electron system. It is found that the phase relaxation rate in two dimensions varies linearly with the temperature as in the more common case of electric field fluctuations, but is proportional to the conductance rather than the resistance. A logarithmic correction factor is found in comparison to an earlier qualitative estimate.     

Quasiclassical Lian-Zuckerman Homotopy Algebras,Courant Algebroids and Gauge Theory

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the A _∞-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A _∞-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by β - γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.     

Quasiclassical Wavefunctions

A convergent quasiclassical formula for the wavefunctions of a closed quantum or wave system is obtained. This is expressed entirely in terms of classical orbits. The result is at the same level as earlier results expressing the spectrum as a finite resurgent sum over composite periodic orbits.     

Autocorrelation of Random Matrix Polynomials

 We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions. Received: 1 August 2002 / Accepted: 25 December 2002 Published online: 7 May 2003 Communicated by P. Sarnak     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.     

Central nucleus-nucleus collisions at relativistic energies with a new method based on Random Matrix Theory

Using the method based on Random Matrix Theory (RMT),the results for the nearest-neighbor distributions obtained from the experimental data on 12C-C collisions at 4.2 AGeV/c have been discussed and compared with the simulated data on 12C-C collisions at 4.2 AGeV/c produced with the aid of the Dubna Cascade Model.The results show that the correlation of secondary particles decreases with an increasing number of charged particles Nch.These observed changes in the nearest-neighbor distributions of charged particles could be associated with the centrality variation of the collisions.