From Gaussian to Gumbel: extreme eigenvalues of complex Ginibre products with exact rates

We consider the product of $k_n$ independent $n\times n$ complex Ginibre matrices and denote its eigenvalues by $Z_1,\dots ,Z_n$. Let $\alpha ={\lim }_{n\to \mathrm{\infty }}n\mathrm{/}{k}_n$. Using the determinantal point process method, we reduce the study of extremal eigenvalues to the evaluation of determinants of certain $n\times n$ matrices. In the modulus case, rotational invariance makes the relevant matrix diagonal, which yields a product representation in terms of Gamma tail probabilities. In the real-part case, the matrix is no longer diagonal; we handle this by a polar-coordinate reduction that introduces an independent uniform angle and leads to explicit formulas involving Gamma variables and trigonometric integrals. After appropriate rescaling, the spectral radius ${\max }_{1\le j\le n}\mathrm{\mid }{Z}_j\mathrm{\mid }$ converges weakly to a nontrivial distribution ${\mathrm{\Phi }}_{\alpha }$ when $\alpha \in (0,+\mathrm{\infty })$, to the Gumbel distribution when $\alpha =+\mathrm{\infty }$, and to the standard normal distribution when $\alpha =0$. The family $\{{\mathrm{\Phi }}_{\alpha }{\}}_{\alpha >0}$ extends continuously to the boundary regimes: ${\mathrm{\Phi }}_{\alpha }$ converges weakly to the standard normal law as $\alpha \to 0^{+}$ and to the Gumbel law as $\alpha \to +\mathrm{\infty }$. Thus the three limiting regimes are connected by the single parameter $\alpha$, yielding a continuous transition from Gaussian to Gumbel distribution. For the spectral radius, we obtain the exact rates of convergence both in the fixed-$\alpha$ regime and at the boundaries $\alpha =0$ and $\alpha =+\mathrm{\infty }$. For the rightmost eigenvalue ${\max }_{1\le j\le n}\mathrm{\Re }{Z}_j$, we establish the convergence rates in the boundary regimes, while for $\alpha \in (0,+\mathrm{\infty })$ we show that the limiting distribution, though not available in closed form, still interpolates continuously between the normal and Gumbel laws.