We consider a Wigner-type ensemble, i.e. large hermitian [Formula: see text] random matrices [Formula: see text] with centered independent entries and with a general matrix of variances [Formula: see text]. The norm of [Formula: see text] is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of [Formula: see text] that substantially improves the earlier bound [Formula: see text] given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields 169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.

中文翻译：

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Wigner 型随机矩阵范数的界限

我们考虑一个Wigner 类型的集合，即大厄米特[公式：见文本] 随机矩阵[公式：见文本]，具有居中的独立条目和一般方差矩阵[公式：见文本]。[公式：见正文]的范数由状态自洽密度的最大支持度渐近给出。我们根据 [公式：见文本] 的幂规范建立了这个最大值的界限，这大大改进了 [O. Ajanki、L. Erdős 和 T. Krüger，通用 Wigner 型矩阵的普遍性，Prob。理论。相对。字段 169 (2017) 667–727]。证明的关键要素是对出现在相应戴森方程的迭代解中的加权 Dyck 路径的贡献进行有效的马尔可夫链近似。