Scaling Limits of Random Graphs from Subcritical Classes

TL;DR

This paper analyzes the scaling limits of uniform random graphs from subcritical classes, showing convergence to the Brownian CRT and providing tail bounds for diameter and height, with applications to outerplanar graphs and first passage percolation.

Contribution

It establishes the convergence of rescaled subcritical random graphs to the Brownian CRT and derives tail bounds, extending understanding of their geometric properties.

Findings

Rescaled graphs converge to the Brownian CRT.

Tail bounds for diameter and height are established.

Results apply to classes like outerplanar graphs and first passage percolation.

Abstract

We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling.