论文标题
随机矩阵产品的焦点的通用性
Universality for cokernels of random matrix products
论文作者
论文摘要
对于随机整数矩阵，$ m_1，\ ldots，m_k \ in \ propatatorName {mat} _n（\ mathbb {z}）$具有独立条目，我们研究了cokernel $ \ operatotorname $ \ operatorname {cok}（cok}（cok}（m_1 \ cdots m_k）$的产品。我们表明，对于一般的矩阵输入分布，此分布将$ n \ to \ infty $收敛为$ n \ to \ infty $，并且更通常显示出$ \ permatatorName {cok}（M_1），\ propatatorNorname {cok}（cok}（m_1m_2）（m_1m_2），\ ldots，\ ldots，\ operatoTareOns的联合分布的普遍限制。此外，我们将普遍的分布表征为将Cohen-Lenstra度量的自然概括与它们之间的Abelian群体序列的自然概括的边际分布，它们之间具有地图，该序列与它们的自动形态数量成正比，使序列与它们的数量成正比。证明将木刻方法的延伸到多组的关节力矩，并依靠与Hall-Littlewood多项式和对称函数身份的联系。作为推论，我们在$ \ mathbb {f} _p $上获得了随机矩阵产品的核心的显式通用分布，因为矩阵大小往往是无穷大的。
For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution converges to a universal one as $n \to \infty$ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of $\operatorname{cok}(M_1),\operatorname{cok}(M_1M_2),\ldots,\operatorname{cok}(M_1 \cdots M_k)$. Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over $\mathbb{F}_p$ as the matrix size tends to infinity.
