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.

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Last updated: 17 Jan 2019