Scaling limits of random graphs from subcritical classes

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 sub-Gaussian tail bounds for the diameter $\mathrm{D}(\mathsf{C}_{n})$ and height $\mathrm{H}(\mathsf{C}_{n}^{\bullet})$ of the rooted random graph $\mathsf{C}_{n}^{\bullet}$. We give analytic expressions for the scaling factor in several cases, including for example the class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_{n}$, where we also show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling.