  In this paper, we first obtain an algebraic formula for the moments of a
centered Wishart matrix, and apply it to obtain new convergence results in the
large dimension limit when both parameters of the distribution tend to infinity
at different speeds. We use this result to investigate APPT (absolute positive
partial transpose) quantum states. We show that the threshold for a bipartite
random induced state on $\C^d=\C^{d_1} \otimes \C^{d_2}$, obtained by partial
tracing a random pure state on $\C^d \otimes \C^s$, being APPT occurs if the
environmental dimension $s$ is of order $s_0=\min(d_1, d_2)^3 \max(d_1, d_2)$.
That is, when $s \geq Cs_0$, such a random induced state is APPT with large
probability, while such a random states is not APPT with large probability when
$s \leq cs_0 $. Besides, we compute effectively $C$ and $c$ and show that it is
possible to replace them by the same sharp transition constant when $\min(d_1,
d_2)^{2}\ll d$.
