Local limits of determinantal processes

TL;DR

This paper investigates the local limits of certain determinantal processes derived from bipartite graphs with diverging degrees, showing they converge to a conditioned Poisson branching process, unifying various models.

Contribution

It establishes the local limit of a broad class of determinantal processes on bipartite graphs with diverging degrees as a conditioned Poisson branching process.

Findings

The local limit is a conditioned Poisson(k) branching process.

Applicable to models like spanning trees, hypertrees, hyperforests, and Grassmannians.

Results cover processes with degrees tending to infinity.

Abstract

Let $H_n$ be the row space of a signed adjacency matrix of a $C_4$-free bipartite bi-regular graph in which one part has degree $d(n)\to\infty$ and the other part has degree $k+1$ where $k\geq 1$ is a fixed integer. We show that the local limit as $n\to \infty$ 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 cell complexes, discrete Grassmanians, incidence matroids and more, as long as their degree tends to $\infty$.