The continuum limit of critical random graphs

Abstract

We consider the Erdős-Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3 , for some fixed λ ∈ R. We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3 , converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.

Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

“…Thus to extend our results to the vacant set problem for general graphs, all one needs is an extension of the refined bounds in (3) to random walks on general graphs. (d) Proof techniques : The techniques used in this paper differ from the standard techniques used to show convergence of such random discrete objects to limiting random tree like metric spaces. One standard technique (used in ) is to construct an exploration process of the discrete object of interest that converges to the exploration process of a continuum random tree (see for beautiful treatments), and encode the “surplus” edges as a random point process falling under the exploration, and show that this point process converges to a Poisson point process in the limit. In this work, we use a different technique that requires less work.…”

“…Remark The Erdős‐Rényi scaling limit identified in can be recovered by taking the limiting random variable to be D er ∼Poisson(1), that is, the scaling limit of ERRG( n −1 + λn −4/3 ) (after rescaling the graph distance by n −1/3 ) is given by (Note that in this case, .) The result for ERRG( n −1 + λn −4/3 ) can be obtained from Theorem by observing the following two facts: (i)The (random) degree sequence of ERRG( n −1 + λn −4/3 ) satisfies Assumption with limiting random variable D er . (ii)Conditional on the event where the degree sequence equals d , ERRG( n −1 + λn −4/3 ) is uniformly distributed over . …”

“…Maximal component sizes in ERRG( n , λ ) were studied extensively in . The scaling limit of the maximal components of ERRG( n , λ ) when viewed as metric spaces was identified in . It is believed that a large class of random discrete structures, in the critical regime, belong to the “Erdős‐Rényi universality class.” Soon after the work , an abstract universality principle was developed in which was used to establish Erdős‐Rényi type scaling limits for a wide array of critical random graph models including the configuration model and various models of inhomogeneous random graphs.…”

“…The scaling limit of the maximal components of ERRG( n , λ ) when viewed as metric spaces was identified in . It is believed that a large class of random discrete structures, in the critical regime, belong to the “Erdős‐Rényi universality class.” Soon after the work , an abstract universality principle was developed in which was used to establish Erdős‐Rényi type scaling limits for a wide array of critical random graph models including the configuration model and various models of inhomogeneous random graphs. It is strongly believed that the components of critical percolation on high‐dimensional tori , and the hypercube also share the Erdős‐Rényi scaling limit, but these problems are open at this point. Vacant set left by random walk (VSRW) on graphs : The second main theme is the area of random interlacements and percolative properties of the vacant set of random walks on finite graphs, see for example, .…”

The diameter of inhomogeneous random graphs

“…Nachmias and Peres obtained the order of the diameter, namely . Addario‐Berry, Broutin, and Goldschmidt proved convergence, in the Gromov–Hausdorff distance, of the rescaled connected components to a sequence of continuous compact metric spaces. In particular, the diameter rescaled by converges in distribution to an absolutely continuous random variable with finite mean.…”