Sanchayan Sen

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

AbstractOne 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):168701, 2003; Wu et al. in Phys Rev Lett 96(14):148702, 2006; Braunstein et al. Int J Bifurc Chaos 17(07):2215–2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006) is as follows: for a wide array of random graph models with degree exponent

Critical window for the configuration model: finite third moment degrees

\tau \in (3,4)

Universality for critical heavy-tailed network models : metric structure of maximal components

τ∈(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

A probabilistic approach to the leader problem in random graphs

n^{(\tau -3)/(\tau -1)}

3

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