Universality for cokernels of random matrix products

Abstract

For random integer matrices M1,…,Mk∈Matn(Z) with independent entries, we study the distribution of the cokernel Cok(M1⋯Mk) of their product. We show that this distribution converges to a universal one as n→∞ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of Cok(M1),Cok(M1M2),…,Cok(M1⋯Mk). 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 Fp as the matrix size tends to infinity.