Random p-adic matrices with fixed zero entries and the Cohen--Lenstra distribution

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.