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Using asymptotics of Toeplitz+Hankel determinants, we establish formulae forthe asymptotics of the moments of the moments of the characteristic polynomialsof random orthogonal and symplectic matrices, as the matrix-size tends toinfinity. Our results are analogous to those that Fahs obtained for randomunitary matrices in [14]. A key feature of the formulae we derive is that thephase transitions in the moments of moments are seen to depend on the symmetrygroup in question in a significant way.

Moments of Moments of the Characteristic Polynomials of Random Orthogonal and Symplectic Matrices