Local limits of determinantal processes

TL;DR

The paper identifies the local weak limit for a broad class of determinantal processes arising from the row-space of signed adjacency matrices on $C_4$-free bipartite bi-regular graphs with high left-degree. The main result shows that the local neighborhood around a uniformly chosen root converges to a variant of a Poisson$(k)$ branching process conditioned to survive, denoted $\\mathbb T_k$, with the limit law depending only on the fixed integer $k$. The authors provide a unified linear-algebraic framework that encompasses models including uniform spanning trees, Kalai's determinantal hypertrees, high-degree cell-complex hyperforests, discrete Grassmannians, and incidence matroids, and they give an explicit distribution for the limit object via a combinatorial quantity $m(I(T),T)$ and automorphism counts. A quenched strengthening and a broad set of corollaries extend previous results (notably NP22 and meszáros2021local) to many high-degree determinantal models, highlighting a deep connection between determinantal measures and local branching structures with potential applications in random combinatorial geometries and matroid theory. The approach blends spectral analysis, transversals in trees, and determinant-based inequalities to control local structure and derive the limiting object.

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 $\infty$.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 1.5
• proof
• Definition 2.1
• Theorem 2.2
• Claim 2.3
• proof