General truncated linear statistics for the top eigenvalues of random matrices

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the β generalised fixed trace Laguerre ensemble. In the original case (β = 2), the purity cumulants are expressed in terms of the large argument expansion of a particular σ-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU(N ) is given in the Appendix.Dedicated to the memory of Santosh Kumar and his work on exact results in RMT and their applications 1

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance and the charge relaxation resistance Rq, where C is the geometric capacitance and Cq the quantum capacitance. We study the limit of a large number of conducting channels N with a Coulomb gas method. We obtain and show that the relative fluctuations are of order 1/N both for Cq and Rq, with strong correlations . The detailed analysis of large deviations involves a second-order phase transition in the Coulomb gas. The two-dimensional phase diagram is obtained.

Chapter 2.6.3 - Ultra-Cold Collisions of Atoms and Molecules

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.

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.

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

Our current understanding of quantum chaos in many-body quantum systems hinges on the random matrix theory (RMT) behavior of eigenstates and their energy level statistics. Although RMT has been remarkably successful in describing "coarse" features of many-body quantum Hamiltonians in chaotic regimes, such as the Wigner-Dyson level spacing statistics or the volume-law behavior of eigenstate entanglement entropy, it remains a challenge to describe their "finer" features, particularly those arising from spatial locality. Here, we show that we can accurately describe the statistical behavior of eigenstate ensembles in many-body Hamiltonians by using pure random states with physical constraints that capture the essential features of the Hamiltonian, specifically spatial locality and symmetries. We demonstrate our approach on local spin Hamiltonians with a scalar U(1) charge. By constructing ensembles of constrained random states that account for two commuting scalar charges playing the role of energy and magnetization, we describe the patterns of entanglement of midspectrum eigenstates beyond their average volume-law behavior, including O(1) corrections and fluctuations, analytically and numerically. When defining the correspondence between quantum chaotic eigenstates in many-body Hamiltonians and RMT ensembles, our work highlights the important role played by spatial locality in describing universal features beyond the volume-law behavior.

Parametrized families of distributions for the circular unitary ensemble in random matrix theory are considered; they are connected to Toeplitz determinants and have many applications in mathematics (for example, to the longest increasing subsequences of random permutations) and physics (for example, to nuclear physics and quantum gravity). We develop a theory for the unknown parameter estimated by an asymptotic maximum likelihood estimator, which, in the limit, behavesas the maximum likelihood estimator if the latter is well defined and the family is sufficiently smooth. They are asymptotically unbiased and normally distributed, where the norming constants are unconventional because of long range dependence.

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erds–Rényi (E–R) random graph is determined by the average degree ⟨k⟩ and p(s) undergoes a dramatic change when ⟨k⟩ is varied around the critical point of the percolation transition, ⟨k⟩ = 1. When ⟨k⟩ ≫ 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at ⟨k⟩ = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein–protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E–R graph.

Multifractality attributed to dual central limit-like convergence effects

Fractal structure of random matrices

We investigate the influence of spin-orbit interaction on ballistic transport through chaotic cavities by using semiclassical methods. Our approach is based on the Landauer formalism and the Fisher-Lee relations, appropriately generalized to spin-orbit interaction, and a semiclassical representation of Green functions. We calculate conductance coefficients by exploiting ergodicity and mixing of suitably combined classical spin-orbit dynamics, and making use of the Sieber-Richter method and its most recent extensions. That way we obtain weak anti-localization and confirm previous results obtained in the symplectic ensemble of Random Matrix Theory.

In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is random matrix from class C 12 [26] or from the following class C 13 of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A n + U n , where Unitary matrix U n from the class C 1 [26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix U n and a nonrandom matrix A n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A n + Ξ n B n , where Ξ n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices. We give the formulation of the Circular Law for random matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is from the so called Class C 12 of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p ( x, y ) of eigenvalues of A n + B n U n ( ε ) C n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p ( x, y ) the Mercury support . If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U n from class C 1 all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U n any diagonal nonrandom matrix A n = ( δ ij a j ) then if the distances between diagonal entries a j are large enough we have almost the same picture of limit distribution of eigenvalues of U n + A n if Unitary matrix U n would be equal to the random matrix Ξ n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ n have the order n −1 . Therefore, the matrix Ξ n looks like orthogonal random matrix. In this paper we use for matrices A n + B n U n ( ε ) C n the triply regularized V -transform[2, 5],( VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays ) where α > 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ n { x,G n ( t, s, γ, ε )} is the normalized spectral function of G -matrix: Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A n + B n U n and the following V - domain G δ , δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A n + B n U n are found: where are eigenvalues of the matrix.

The mechanical action of light on atoms is nowadays a tool used ubiquitously in cold atom physics. In the semiclassical regime where the atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires in the most general case numerical approaches. Here we show that this problem can be tackled in a pure analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case where the atom is simultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors, and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.