Universal constant order fluctuations for the cokernels of block triangular matrices

TL;DR

This work extends the universality of cokernel fluctuations from matrix products to a broad class of random block lower triangular matrices. By developing a Hom-moment approach and a careful decomposition over subgroup chains, the authors show that the Sylow $p$-subgroups of the cokernels exhibit constant-order fluctuations in the same universality class as matrix products studied by Nguyen–Van Peski, with limiting moments $\frac{c(G,\ell(G))}{\ell(G)!}$. They provide detailed proofs (via a decomposition by $w(\mathbf{g})$ and $t(\mathbf{g})$) and, in a subsequent, streamlined argument, weaken the growth condition on the number of factors from previous results, requiring only $\lim_{n\to\infty}\frac{\log k_n}{n}=0$. The results reinforce the robustness of cokernel universality across structured random matrix ensembles and connect the block-model fluctuations to the established matrix-product framework via the rescaled moment method and the limiting object $\mathcal{L}_{d,p^{-1},\chi}$.

Abstract

We prove that for a large class of random block lower triangular matrices, the Sylow $p$-subgroups of their cokernels have the same constant order fluctuations as that of the matrix products studied by Nguyen and Van Peski in arXiv:2409.03099. We also show that the theorem of Nguyen and Van Peski remains true under a weaker assumption on the number of factors in the matrix products.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Nguyen and Van Peski nguyen2024rank
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Theorem 2.4
• Claim 3.1