论文标题
随机$ p $ -adic Hermitian矩阵的焦点的普遍性
Universality of the cokernels of random $p$-adic Hermitian matrices
论文作者
论文摘要
在本文中，我们研究了整数$ \ natercal {o} $ a extratic Extension $ k $ of $ \ mathbb {q} _p $的cokernel的分布。对于每个正整数$ n $，让$ x_n $成为一个随机的$ n \ times n $ hermitian矩阵，超过$ \ nathcal {o} $，其上三角式条目是独立的，其降低并不集中在某些值上。我们表明，$ x_n $的cokernel的分布总是收敛到同一分布，这不取决于$ x_n $的选择，即$ n \ rightarrow \ infty \ infty $，并为限制分配提供了明确的公式。在ICM 2022讲座的情况下，这是$ \ mathbb {q} _p $的二次扩展名的木环。
In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers $\mathcal{O}$ of a quadratic extension $K$ of $\mathbb{Q}_p$. For each positive integer $n$, let $X_n$ be a random $n \times n$ Hermitian matrix over $\mathcal{O}$ whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of $X_n$ always converges to the same distribution which does not depend on the choices of $X_n$ as $n \rightarrow \infty$ and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood in the case of the ring of integers of a quadratic extension of $\mathbb{Q}_p$.
