Even-power characteristic polynomial moments for Gaussian, Laguerre, and Jacobi beta ensembles are shown to have large-N expansions with explicit error bounds, a universal Barnes-G constant, and new formulas for weight parameters proportional to N.
Benford's law and the C$\beta$E
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abstract
We study the individual digits for the absolute value of the characteristic polynomial for the Circular $\beta$-Ensemble. We show that, in the large $N$ limit, the first digits obey Benford's Law and the further digits become uniformly distributed. Key to the proofs is a bound on the rate of convergence in total variation norm in the CLT for the logarithm of the absolute value of the characteristic polynomial.
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Moments of characteristic polynomials for classical $\beta$ ensembles
Even-power characteristic polynomial moments for Gaussian, Laguerre, and Jacobi beta ensembles are shown to have large-N expansions with explicit error bounds, a universal Barnes-G constant, and new formulas for weight parameters proportional to N.