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We study moments of the logarithmic derivative of characteristic polynomialsof orthogonal and symplectic random matrices. In particular, we compute theasymptotics for large matrix size, $N$, of these moments evaluated at pointswhich are approaching 1. This follows work of Bailey, Bettin, Blower, Conrey,Prokhorov, Rubinstein and Snaith where they compute these asymptotics in thecase of unitary random matrices.

Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$