On the moments of the characteristic polynomial of a Ginibre random matrix

Webb, Christian 
Wong, Mo Dick 

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In this article we study the large N asymptotics of complex moments of the absolute value of the characteristic polynomial of a N×N complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large N asymptotics of E|det(GN−z)|γ, where GN is a N×N matrix whose entries are i.i.d and distributed as N−1/2Z, Z being a standard complex Gaussian, ℜ(γ)>−2, and |z|<1. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: det(∫Cwiw―j|w−z|γe−N|w|2d2w)i,j=0N−1. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight |w−z|γe−N|w|2d2w along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.

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4901 Applied Mathematics, 4902 Mathematical Physics, 49 Mathematical Sciences
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Proceedings of the London Mathematical Society
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Wiley Online
Engineering and Physical Sciences Research Council (EP/L016516/1)
Christian Webb was supported by the Academy of Finland grants 288318 and 308123. Mo Dick Wong is supported by the Croucher Foundation Scholarship and EPSRC grant EP/L016516/1 for his PhD study at Cambridge Centre for Analysis.