2024/09/02 by Dong Yeap Kang, Jungin Lee, Kang, Dong Yeap +3 · 1 citation
Mathematics · Computer Science · #advanced mathematical theories #Topological and Geometric Data Analysis #Random Matrices and Applications
paper · pdf · doi:10.1007/s00029-026-01183-5
In this paper, we study the distribution of the cokernels of random p-adic matrices with fixed zero entries. Let Xn be a random n × n matrix over ℤp in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable ξ∈ ℤp. We consider the minimal number of random entries of Xn required for the cokernel of Xn to converge to the Cohen--Lenstra distribution. When ξ is given by the Haar measure, we prove a lower bound of the number of random entries and prove its converse-type result using random regular bipartite multigraphs. When ξ is a general random variable, we determine the minimal number of random entries. Let Mn be a random n × n matrix over ℤp with k-step stairs of zeros and the other entries given by independent random ε-balanced variables valued in ℤp. We prove that the cokernel of Mn converges to the Cohen--Lenstra distribution under a mild assumption. This extends Wood's universality theorem on random p-adic matrices.