The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

Abstract

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16) (3,4), distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n (τ −3)/(τ −1) . In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes "without replacement" (Aldous and Limic Electron in J Probab 3(3):59, 1998), yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent τ ∈ (3, 4) where edges are rescaled by n −(τ −3)/(τ −1) yielding the first rigorous proof of the above conjecture. in this case are compact "tree-like" random fractals with a dense collection of hubs (infinite degree vertices), a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal (τ − 2)/(τ − 3) a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent τ ∈ (3, 4). Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in Athreya et al. (2014) and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound.:Keywords Multiplicative coalescent · p-trees · Inhomogeneous continuum random trees · Critical random graphs · Gromov-Hausdorff distance · Gromov-weak topology Mathematics Subject Classification Primary 60C05 · 05C80List of notation and terminology (S t Critical exponentsSpace of counting measures on R + equipped with the vague topologyMinimal number of open balls with radius δ required to cover a metric space M For a tree t ∈ T * I,(k+ ) , the functional defined in (4.24) ht(t)Height of a tree t with edge lengths incorporated into the distanceRoot-to-vertex weights and measures in T θ (∞) defined in (2.7) and (2.8). See ...

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

“…Then the maximal components in the critical scaling window still belong to the Erdős‐Rényi universality class as in () above with distances scaling like n −1/3 . This contrasts drastically with critical percolation on these random graphs where maximal components with distances scaled by converge to limiting random fractals .…”

“…Personal communication from Remco van der Hofstad suggests that the limits here differ substantially from critical percolation. Scaling limits for critical percolation in such heavy tailed random graphs were derived in , where scaling limits of maximal components in Aldous's multiplicative coalescent were established in terms of tilted inhomogeneous continuum random trees. One ramification of these results ([, Thm.…”

“…Scaling limits for critical percolation in such heavy tailed random graphs were derived in , where scaling limits of maximal components in Aldous's multiplicative coalescent were established in terms of tilted inhomogeneous continuum random trees. One ramification of these results ([, Thm. 1.2]) is the continuum scaling limits of the maximal components in the critical regime of the so‐called Norros‐Reittu model where the driving weight sequence is assumed to have heavy tails with exponent τ ∈ (3,4).…”

“…The first space T IJ was formulated in where it was used to study trees spanning a finite number of random points sampled from an inhomogeneous continuum random tree (ICRT). A more general space was used in the proofs in . The index I in T IJ and is needed for the purpose of keeping track of the number of marked “hubs,” that is, vertices of high (or infinite) degrees in such trees (see for a proper definition).…”

“…Proof The result is essentially contained in [, Lem. 4.9], and we only outline how to extract the result from its proof.…”