矩阵乘积的秩波动与生长群的矩方法
Rank fluctuations of matrix products and a moment method for growing groups
AI总结：
研究随机整数矩阵乘积的余核统计，揭示其在大规模极限下的普遍收敛性，提出适应生长群的矩方法理论。
摘要：
我们考虑由独立n×n随机整数矩阵乘积生成的余核G_n，在n和k同时趋于无穷的 regime 中，证明余核统计普遍收敛于反射泊松海，一个由arXiv:2312.11702构建的相互作用粒子系统。特别地，A_k...A_2A_1 mod p的corank近似为log_p k，其波动为O(1)并收敛于arXiv:2310.12275定义的离散随机变量。与以往研究不同，G_n不收敛于随机有限群；例如，G_n的p-秩发散。通常的随机群矩方法不适用，本文提出适应生长群的矩方法定理，证明在特定条件满足下，p-秩及其他统计量的波动仍收敛于极限随机变量。
英文摘要：
We consider the cokernel $G_n = \mathbf{Cok}(A_{k} \cdots A_2 A_1)$ of a product of independent $n \times n$ random integer matrices with iid entries from generic nondegenerate distributions, in the regime where both $n$ and $k$ are sent to $\infty$ simultaneously. In this regime we show that the cokernel statistics converge universally to the reflecting Poisson sea, an interacting particle system constructed in arXiv:2312.11702, at the level of $1$-point marginals. In particular, $\operatorname{corank}(A_{k} \cdots A_2 A_1 \pmod{p}) \sim \log_p k$, and its fluctuations are $O(1)$ and converge to a discrete random variable defined in arXiv:2310.12275. The main difference with previous works studying cokernels of random matrices is that $G_n$ does not converge to a random finite group; for instance, the $p$-rank of $G_n$ diverges. This means that the usual moment method for random groups does not apply. Instead, we proceed by proving a `rescaled moment method' theorem applicable to a general sequence of random groups of growing size. This result establishes that fluctuations of $p$-ranks and other statistics still converge to limit random variables, provided that certain rescaled moments $\mathbb{E}[\#\operatorname{Hom}(G_n,H)]/C(n,H)$ converge.
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