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One major open conjecture in the area of critical random graphs, formulatedby statistical physicists, and supported by a large amount of numericalevidence over the last decade [23, 24, 28, 63] is as follows: for a wide arrayof random graph models with degree exponent $\tau\in (3,4)$, distances betweentypical points both within maximal components in the critical regime as well ason the minimal spanning tree on the giant component in the supercritical regimescale like $n^{(\tau-3)/(\tau-1)}$.  In this paper we study the metric space structure of maximal components ofthe multiplicative coalescent, in the regime where the sizes converge toexcursions of L\'evy processes "without replacement" [10], yielding acompletely new class of limiting random metric spaces. A by-product of theanalysis yields the continuum scaling limit of one fundamental class of randomgraph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by$n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the aboveconjecture. The limits in this case are compact "tree-like" random fractalswith finite fractal dimensions and with a dense collection of hubs (infinitedegree vertices) a finite number of which are identified with leaves to formshortcuts. In a special case, we show that the Minkowski dimension of thelimiting spaces equal $(\tau-2)/(\tau-3)$ a.s., in stark contrast to theErd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It isgenerally believed that dynamic versions of a number of fundamental randomgraph models, as one moves from the barely subcritical to the critical regimecan be approximated by the multiplicative coalescent. In work in progress, thegeneral theory developed in this paper is used to prove analogous limit resultsfor other random graph models with degree exponent $\tau\in (3,4)$.

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