Cohen–Lenstra Distribution for Sparse Matrices with Determinantal Biasing

Mészáros, András (2025) Cohen–Lenstra Distribution for Sparse Matrices with Determinantal Biasing. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2025 (3). ISSN 1073-7928

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Abstract

Let us consider the following matrix . The columns of are indexed with and the rows are indexed with . The row corresponding to is given by , where is the standard basis of . Let be random submatrix of , where the probability that we choose a submatrix is proportional to . Let be a prime. We prove that the asymptotic distribution of the -Sylow subgroup of the cokernel of is given by the Cohen–Lenstra heuristics. Our result is motivated by the conjecture that the first homology group of a random two dimensional hypertree is also Cohen–Lenstra distributed.

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