Simplex links in determinantal hypertrees

TL;DR

This paper provides a detailed probabilistic and structural analysis of higher-dimensional spanning trees in simplices, revealing new distributional properties and applications in topology, graph spectra, and expander theory.

Contribution

It introduces an inductive description of determinantal probability measures for higher-dimensional spanning trees and characterizes links and forests in simplicial complexes.

Findings

Derived marginal distributions of simplex links in random trees

Characterized higher-dimensional spanning trees of simplicial cones

Proved property (T) for fundamental groups of unions of determinantal 2-trees

Abstract

We deduce a structurally inductive description of the determinantal probability measure associated with Kalai's celebrated enumeration result for higher--dimensional spanning trees of the $n-1$--simplex. As a consequence, we derive the marginal distributions of the simplex links in such random trees. Along the way, we also characterize the higher--dimensional spanning trees of every other simplicial cone in terms of the higher--dimensional rooted forests of the underlying simplicial complex. We also apply these new results to random topology, the spectral analysis of random graphs, and the theory of high dimensional expanders. One particularly interesting corollary of these results is that the fundamental group of a union of $o(\log n)$ determinantal 2--trees has Kazhdan's property (T) with high probability.