Random integral matrices: universality of surjectivity and the cokernel

Local and global universality of random matrix cokernels

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of all these models. In particular, we find the probability that a sandpile group of an Erdos-Renyi random graph is cyclic, answering a question of Lorenzini from 2008.

矩阵对称性下的余核量化 universality

Quantative universality for cokernels of matrices with symmetries

本文通过量化方法证明了具有对称性的随机整数矩阵余核的 universality，解决了伍德关于有效收敛速率的问题。

经典随机矩阵模型在$p$-adic整数上的余核 universality 的尖锐阈值

Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers

本文证明了$\frac{\log n}{n}$是经典随机矩阵模型在$p$-adic整数上余核分布universality的尖锐阈值，改进了之前的结果，并应用于沙堆群的泛化。

取值于戴德金整环的随机矩阵余核的普适性

Universality for Cokernels of Dedekind Domain Valued Random Matrices

该研究利用Wood的矩方法，针对元素取值于带有限商的可数戴德金整环的随机矩阵，提出并证明了一个可得到余核分布普适性结果的条件，拓展了此前整数矩阵场景下的相关结论。

有限域上随机矩阵余核普适性的尖锐阈值

Sharp threshold for universality of cokernels of random matrices over finite fields

该研究确定有限域上随机矩阵余核普适性的尖锐阈值，证明满足特定元素概率条件的随机矩阵余核依分布收敛到均匀随机矩阵余核的极限律，解答了Wood的公开问题。

具有非均匀平衡列的随机 $p$-adic 矩阵的余核的普适性

Universality of the cokernels of random $p$-adic matrices with inhomogeneously balanced columns

本文证明了当列平衡参数满足一定条件时，随机 $p$-adic 矩阵的余核分布收敛到 Haar 随机矩阵的极限分布，将 Nguyen 和 Wood 的普适性定理推广到非均匀平衡列情形。