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

In this article, we study the large N asymptotics of complex moments of the absolute value of the characteristic polynomial of an 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(G_N − z)|^γ, where G_N is an N × N matrix whose entries are i.i.d. and distributed as N^−1/2Z, Z being a standard complex Gaussian, Re(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(∫_C wiwj |w − z|^γe^−N|w|2 d² i,j=0.. 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|2 d² walong 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.