Random matrices over the p-adic integers have cokernels — quotient groups that encode the matrix's algebraic structure. As the matrix size n grows, these cokernels converge to a universal distribution that depends only on the symmetry class (non-symmetric, symmetric, or alternating) and not on the specific entry distribution. This is the p-adic analog of the Wigner semicircle law: local details wash out at large scales.
Jung, Lee, and Yu (arXiv:2603.12879) identify the exact threshold at which this universality kicks in. For α_n-balanced random matrices — where each entry has probability at least α_n of taking any particular value mod p — universality holds when α_n ≫ (log n)/n and fails when α_n ~ (log n)/n. The threshold is (log n)/n, sharp to leading order.
The sharpness is the result. Below the threshold, the matrix retains enough memory of its entry distribution to produce non-universal cokernels. Above the threshold, mixing is sufficient to erase this memory. At the threshold itself (c = 1 in their parametrization), the phase transition occurs: universality breaks. The transition is not gradual — there's a specific scale at which the algebraic structure of the cokernel forgets its origins.
The application connects to graph theory: sandpile groups of random graphs are cokernels of Laplacian matrices. The universality threshold for cokernels translates directly to a threshold for when sandpile group statistics become universal across different random graph models. The unified framework treats all symmetry types simultaneously rather than case by case — the same threshold governs non-symmetric, symmetric, and alternating matrices.