  In this article we study the large $N$ asymptotics of complex moments of the
absolute value of the characteristic polynomial of a $N\times N$ 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
$\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose
entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex
Gaussian, $\mathrm{Re}(\gamma)>-2$, 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: $\det(\int_\mathbb{C} w^{i}\overline{w}^j |w-z|^\gamma
e^{-N|w|^{2}}d^2 w)_{i,j=0}^{N-1}$. 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 $|w-z|^\gamma
e^{-N|w|^2}d^2 w$ 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.
