Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble

Abstract

The statistics of the smallest eigenvalue of Wishart-Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of Wishart-Laguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices ($\beta =1$) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices ($\beta =2$). In the present work we extend this to $\beta$-Wishart-Laguerre ensembles for the case when exponent $\alpha$ in the associated Laguerre weight function, ${\lambda }^{\alpha }{e}^{-\beta \lambda /2}$, is a non-negative integer, while $\beta$ is positive real. This also gives access to the smallest eigenvalue density of fixed trace $\beta$-Wishart-Laguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of $n$ (the matrix dimension) and $\alpha$ also enable us to compare with Tracy-Widom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the Wigner-Smith matrix and are relevant to the problem of quantum chaotic scattering.

Published version

References in corpus (16)

• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Nonlinear statistics of quantum transport in chaotic cavities
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
• A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
• Eigenvalue distributions for some correlated complex sample covariance matrices
• Efficient semiclassical approach for time delays
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard
• Smallest eigenvalue density for regular or fixed-trace complex Wishart-Laguerre ensemble and entanglement in coupled kicked tops
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Effect of chiral symmetry on chaotic scattering from Majorana zero modes
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

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• Robust non-ergodicity of ground state in the $\beta$ ensemble
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance