Local limits of determinantal processes

2025/10/22 by Asaf Nachmias, Nachmias, Asaf, Yuval Peled +1
Mathematics · #Random Matrices and Applications #Advanced Combinatorial Mathematics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2510.19563

Abstract

Let H_n be the row space of a signed adjacency matrix of a C₄-free bipartite bi-regular graph in which one part has degree d(n)→∞ and the other part has degree k+1 where k≥ 1 is a fixed integer. We show that the local limit as n→ ∞ of the determinantal process corresponding to the orthogonal projection on H_n is a variant of a Poisson(k) branching process conditioned to survive. This setup covers a wide class of determinantal processes such as uniform spanning trees, Kalai's determinantal hypertrees, hyperforests in regular polytopal complexes, discrete Grassmanians and incidence matroids, as long as their degree tends to ∞.

Citations

• Using dense graph limit theory to count cocycles of random simplicial complexes
• The homology torsion growth of determinantal hypertrees
• The 2-torsion of determinantal hypertrees is not Cohen-Lenstra
• Bounds on the mod 2 homology of random 2-dimensional determinantal hypertrees
• Coboundary expansion for the union of determinantal hypertrees
• Topological expansion for posets and the homological k-connectivity of random q-complexes
• Simplex links in determinantal hypertrees
• The local limit of uniform spanning trees
• Enumeration and randomized constructions of hypertrees
• Recurrence of Distributional Limits of Finite Planar Graphs

Related

• The local limit of uniform spanning trees
• Determinantal random subgraphs
• Multipartite random graphs with given degrees: local limit, revisiting the giant, distances
• Local limit of the random degree constrained process
• Hyperdeterminantal point processes