Giants through higher-order paths in random simplicial complexes

TL;DR

This paper studies the emergence and properties of giant components in high-dimensional random simplicial complexes, revealing phase transitions and structural behaviors using local-weak convergence techniques.

Contribution

It characterizes the phase transition of the largest d-dimensional component in MRSCs and identifies a discontinuous vertex phase transition in Linial-Meshulam complexes.

Findings

In the subcritical regime, the largest component has Θ(log n) simplices.

In the supercritical regime, the proportion of 1-simplices in the giant is explicitly determined.

Vertices in the giant component undergo a discontinuous phase transition from 0 to 1.

Abstract

We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between $(d-1)$- and $d$-dimensional simplices. The phase transition of the largest $d$-dimensional connected component is determined in terms of the parameter $\lambda$ that governs the number of $d$-simplices incident to a typical $(d-1)$-simplex. In the subcritical regime, we show that the largest component contains $\Theta(\log n)$ many $(d-1)$-simplices with high probability in the MRSC model. In the supercritical regime, we determine the asymptotic proportion of $1$-simplices in the giant component in dimension $2$, for $\lambda_c < \lambda < \bar{\lambda}$, where $\bar{\lambda} > 4$ is an explicit constant. In particular, for Linial-Meshulam complexes, this result holds throughout the entire supercritical regime. Additionally, we show that the number of vertices in the giant component undergoes a discontinuous phase transition in $d$-dimensional Linial-Meshulam complexes, in the sense that the asymptotic proportion of vertices in the giant jumps from $0$ to $1$. Our approach is based on local-weak convergence. We establish local-weak convergence in probability for the MRSC model and prove the concentration result via a refined analysis of the breadth-first exploration process, which tracks contributions from newly discovered and previously explored vertices.