Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)

TL;DR

This paper explores the geometric and metric properties of the intersection lattice of discriminantal arrangements, revealing hypercube embeddings and median structures, and analyzes probabilistic thresholds related to circuit overlaps.

Contribution

It demonstrates that the intersection lattice forms a median graph embedded in a hypercube and establishes probabilistic thresholds for circuit overlaps.

Findings

The cover graph is isometrically embedded into a hypercube.

The intersection lattice is a median graph with Hamming distance.

A Poisson limit and sharp threshold are proved for overlaps.

Abstract

We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making it a partial cube and a median graph, with distances given by the Hamming distance and geodesics described by symmetric differences. We also prove a Poisson limit and a sharp threshold for overlaps of random circuit families, revealing an underlying hypercube geometry.