AI 中文总结
基于结式分布方法，证明了随机p进多项式零点及随机p进矩阵特征多项式的普适性，扩展了Haar模型公式，揭示了相关模型的鲁棒性。
AI 中文摘要
我们证明了具有独立、足够非集中系数的随机p进多项式零点的普适性结果。我们表明，当次数趋于无穷时，\u211a_p的任意有限扩域中的极限联合根统计量是普适的，且与Haar系数模型的极限根统计量一致，后者的极限根统计量由Caruso确定（arXiv:2110.03942）。特别地，我们的结果将Haar模型公式扩展到广泛的系数分布类和\u211a_p有限扩域上的一般联合根统计量。我们的方法基于一种新方法，我们称之为结式分布方法。我们不直接研究随机多项式的根，而是分析其与合适固定测试多项式的结式赋值的分布。我们证明这些结式分布决定了多项式的极限律，结合合适的次数估计，可得到一般根统计量的收敛性。作为进一步应用，我们将相同方法应用于具有独立元素的随机p进矩阵的特征多项式，证明特征多项式的 distinguished 因子收敛于与元素分布无关的普适极限分布，从而表明此类随机矩阵模型具有显著的鲁棒性。
英文摘要
We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.
Comments46 pages. Major revision with substantial reorganization and rewriting throughout. The only substantive technical update is the addition of the missing moment-tightness argument for random matrices, completing the random matrix eigenvalue universality theorem. All other changes are organizational or expository. Comments welcome!
