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Differential Identities for the Structure Function of Some Random Matrix Ensembles

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Abstract

The structure function of a random matrix ensemble can be specified in terms of the covariance of the linear statistics \(\sum _{j=1}^N e^{i k_1 \lambda _j}\), \(\sum _{j=1}^N e^{-i k_2 \lambda _j}\) for Hermitian matrices, and the same with the eigenvalues \(\lambda _j\) replaced by the eigenangles \(\theta _j\) for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation \(\rho _{(2)}\). For the circular \(\beta \)-ensemble of unitary matrices, and with \(\beta \) even, we characterise the bulk scaling limit of \(\rho _{(2)}\) as the solution of a linear differential equation of order \(\beta + 1\)—a duality relates \(\rho _{(2)}\) with \(\beta \) replaced by \(4/\beta \) to the same equation. Asymptotics obtained in the case \(\beta = 6\) from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in \(\beta /2\) which determines the coefficient of \(|k|^{11}\) in the small |k| expansion of the structure function for general \(\beta > 0\). For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Brézin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.

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Notes

  1. Duy Khanh Trinh (personal correspondence) has pointed out that the normalising factor \((-\kappa )^k\) on the RHS should be deleted and replaced by a factor of \(-(1/\kappa )\).

  2. The package Mathematica can verify this, but was unable to integrate the LHS independently.

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Acknowledgements

This research is part of the program of study supported by the Australian Research Council Centre of Excellence ACEMS, and the Discovery Project grant DP210102887.

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  1. School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical & Statistical Frontiers, University of Melbourne, Victoria, 3010, Australia

    Peter J. Forrester

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Correspondence to Peter J. Forrester.

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Communicated by Giulio Biroli.

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Forrester, P.J. Differential Identities for the Structure Function of Some Random Matrix Ensembles. J Stat Phys 183, 33 (2021). https://doi.org/10.1007/s10955-021-02767-5

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