2024/01/23 by Alexander Van Werde, Van Werde, Alexander · 1 citation
Mathematics · #05C50 #15B52 #60B20 #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2401.12655
openalex publication_date 2024/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The walk matrix associated to an n× n integer matrix X and an integer vector b is defined by W := (b,X b, . . . ,Xn-1 b). We study limiting laws for the cokernel of W in the scenario where X is a random matrix with independent entries and b is deterministic. Our first main result provides a formula for the distribution of the pm-torsion part of the cokernel, as a group, when X has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ℤ[x]-module structure. The motivation for this work arises from an open problem in spectral graph theory which asks to show that random graphs are often determined up to isomorphism by their (generalized) spectrum. Sufficient conditions for generalized spectral determinacy can namely be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper