The aim of this talk is to give an overview of some recent results in two interconnected areas:

a) Random discrete structures: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on $n$ vertices and degree exponent $\tau>3$, typical distance 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 $n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.

More generally, recent research has provided strong evidence to believe that several objects, including (i) components under critical percolation, (ii) the vacant set left by a random walk, and (iii) the minimal spanning tree, constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed at a complete resolution of these conjectures.

b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90’s, the proof of which relies on a variation of Stein’s method and a quantification of a classical argument in percolation theory.

Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.