**Self-similar real trees defined as fixed-points and their geometric properties.**N. Broutin and H. Sulzbach. Submitted (47 p.), 2016. [arXiv:1610.05331] [±]

We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence, the uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [*Ann Probab*, vol. 39, 2011]. The method happens to be especially powerful to treat cases where the natural mass measure on the real tree only provides weak estimates on the Hausdorff dimension.

**Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph.**S. Bhamidi, N. Broutin, S. Sen, and X. Wang. Submitted (99 p.), 2014. [arXiv:1411.3417] [±]

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter $t$ (usually related to edge density) and a (model dependent) critical time tc which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdős--Rényi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like $n^{2/3}$) and (b) the structure of components (rescaled by $n^{−1/3}$) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical Erdős-Rényi random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions are the same as the Erdős-Rényi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental ran- dom graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erdős-Rényi random graph. As a by product we also get results for component sizes at criticality for a general class of inhomogeneous random graphs.

**Reversing the cut tree of the Brownian continuum random tree.**N. Broutin and M. Wang. Submitted (24 p.), 2014. [arXiv:1408.2924] [±]

Consider the logging process of the Brownian continuum random tree (CRT)
$\cal T$ using a Poisson point process of cuts on its skeleton
[Aldous and Pitman, *Ann. Probab.*, vol. 26, pp. 1703--1726, 1998].
Then, the cut tree introduced by Bertoin and Miermont describes the genealogy
of the fragmentation of $\cal T$ into connected components
[*Ann. Appl. Probab.*, vol. 23, pp. 1469--1493, 2013].
This cut-tree $\operatorname{cut}(\cal T)$ is distributed as another Brownian CRT,
and is a function of the original tree $\cal T$ and of the randomness in the logging process.
We are interested in reversing the transformation of $\cal T$ into $\operatorname{cut}(\cal T)$:
we define a *shuffling* operation, which given a Brownian CRT $\cal H$, yields
another one $\operatorname{shuff}(\cal H)$ distributed in such a way that $(\cal T,
\operatorname{cut}(\cal T))$ and $(\operatorname{shuff}(\cal H), \cal H)$ have the same distribution.