The dimension of sparse and co-sparse random graph orders
Pu Gao, Arnav Kumar
TL;DR
The paper investigates the dimension of random graph orders generated from Erdős–Rényi and random bipartite graphs via transitive closure, with a focus on the sparse regime $p=O(1/n)$ and the co-sparse regime near $p=1$. It develops general, decomposition-based upper bounds on poset dimension and applies them to $P_{\mathcal{B}(n,n,p)}$ and $P_{\mathcal{G}(n,p)}$, obtaining precise asymptotics and bounds in the sparse and co-sparse ranges. Surprisingly, there is no abrupt jump in dimension at the conjectured phase transition $p\approx 1/n$; instead, the results show gradual growth bounds, e.g., $\dim P_{\mathcal{B}(n,n,p=c/n)}$ lies between constants times $c/\log c$ and $c\log^2 c$, while $\dim P_{\mathcal{G}(n,p=c/n)}$ exhibits exponential growth in $c$ within the sparse window. The authors also connect these findings to causal-set theory and provide a general upper bound on poset dimension based on suborder decompositions, which is of independent interest for sparse-graph induced posets.
Abstract
A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$ were previously studied when $p=Ω(\log n/n)$ and when $p$ is not too close to 1. There is a conjectured phase transition in the sparse range at $p=1/n$. In this paper, we investigate this conjectured phase transition and estimate the dimension of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ when $p=O(1/n)$. For the random bipartite order, we additionally estimate its dimension in the co-sparse regime, thereby closing all previously open ranges of $p$. Finally, we establish a general upper bound on the dimension of partial orders based on their decompositions into suborders, a result that is of independent interest.