2023/03/16 by Cheong, Gilyoung, Yu, Myungjun · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)
paper · doi:10.48550/arxiv.2303.09125
Given a prime p, let P(t) be a non-constant monic polynomial in t over the ring ℤp of p-adic integers. Let Xn be an n × n random matrix over ℤp with independent entries that lie in any residue class modulo p with probability at most 1 - ε for a fixed real number 0 < ε< 1. We prove that as n → ∞, the distribution of the cokernel cok(P(Xn)) of P(Xn) converges to the distribution given by a finite product of some explicit measures that resemble Cohen--Lenstra measures. For example, the random matrix Xn can be taken as a Haar-random matrix or a uniformly random (0,1)-matrix. We consider the distribution of cok(P(Xn)) as a distribution of modules over ℤp[t]/(P(t)), which gives us a clearer formulation in comparison to considering the distribution as that of abelian groups. For the proof, we first reduce our problem into a problem over ℤ/pkℤ, for large enough positive integer k, in place of ℤp. Then we use a result of Sawin and Wood to reduce our problem into another problem of computing the limit of the expected number of surjective (ℤ/pkℤ)[t]/(P(t))-linear maps from cok(P(Xn)) modulo pk to a fixed finite size (ℤ/pkℤ)[t]/(P(t))-module G. To estimate the expected number and compute the desired limit, we carefully adopt subtle techniques developed by Wood, which were originally used to compute the asymptotic distribution of the p-part of the sandpile group of a random graph.