{"ID":2858008,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.07942","arxiv_id":"2510.07942","title":"Precise convergence rate of spectral radius of product of complex Ginibre","abstract":"Let $Z_1, \\cdots, Z_n$ denote the eigenvalues of the product $\\prod_{j=1}^{k_n} \\boldsymbol{A}_j$, where $\\{\\boldsymbol{A}_j\\}_{1 \\le j \\le k_n}$ are independent $n\\times n$ complex Ginibre matrices. Define $α= \\lim\\limits_{n \\to \\infty} \\frac{n}{k_n}$. We prove that $X_n,$ a suitably rescaled version of $\\max_{1 \\le j \\le n} |Z_j|^2,$ converges weakly as follows: to a non-trivial distribution $Φ_α$ for $α\\in (0, +\\infty)$, to the Gumbel distribution when $α= +\\infty$, and to the standard normal distribution when $α= 0$. This result reveals a phase transition at the boundaries of $α$. Furthermore, we establish the exact rates of convergence in each regime.","short_abstract":"Let $Z_1, \\cdots, Z_n$ denote the eigenvalues of the product $\\prod_{j=1}^{k_n} \\boldsymbol{A}_j$, where $\\{\\boldsymbol{A}_j\\}_{1 \\le j \\le k_n}$ are independent $n\\times n$ complex Ginibre matrices. Define $α= \\lim\\limits_{n \\to \\infty} \\frac{n}{k_n}$. We prove that $X_n,$ a suitably rescaled version of $\\max_{1 \\le j...","url_abs":"https://arxiv.org/abs/2510.07942","url_pdf":"https://arxiv.org/pdf/2510.07942v4","authors":"[\"Yutao Ma\",\"Xujia Meng\"]","published":"2025-10-09T08:39:21Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}