**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.

**And/or trees: a local limit point of view.**N. Broutin and C. Mailler. Submitted (37 pages), 2015. [arXiv:1510.06691] [±]

We present here a new and universal approach for the study of random and/or trees,
unifying in one framework many different models, including some novel models,
not yet understood in the literature.
An and/or tree is a Boolean expression represented in (one of) its tree shape.
Fix an integer $k$, take a sequence of random (rooted) trees of increasing sizes, say
$(t_n)_{n\ge 1}$, and label each of these random trees uniformly at random in order
to get a random Boolean expression on $k$ variables.

We prove that, under rather weak local conditions on the sequence of random trees
$(t_n)_{n\ge 1}$, the distribution induced on Boolean functions by this procedure
converges as $n\to\infty$. In particular, we characterise two different behaviours
of this limit distribution depending on the shape of the local limit of
$(t_n)_{n\ge 1}$: a *degenerate* case when the local limit has no leaves;
and a non degenerate case, which we are able to describe in more details under
stronger but reasonable conditions. In this latter case, we provide a relationship
between the probability of a given Boolean function and its *complexity*.
The examples we cover include, in a unified way, trees that interpolate between
models with logarithmic typical distances (such as random binary search trees) and
other ones with square root typical distances (such as conditioned Galton--Watson trees).

**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.

**Bounded monochromatic components for random graphs.**N. Broutin and R.J. Kang. Submitted (20 pages), 2014. [arXiv:1407.3555] [±]

We consider vertex partitions of the binomial random graph $G_{n,p}$.
For $np\to\infty$, we observe the following phenomenon: for any partition into
asymptotically fewer than $\chi(G_{n,p})$ parts, i.e. $o(np/\log np)$ parts, there
must be one part whose induced subgraph has a connected component of order at least
roughly the average part size.
Stated another way, we consider the *$t$-component chromatic number*, the
smallest number of colours needed in a colouring of the vertices such that each
colour class induces a subgraph of maximum component order at most $t$.
As long as $np \to \infty$, there is a threshold around $t = \Theta(\log_b np)$
(where $b = 1/(1-p)$), such that if $t$ is smaller then the $t$-component chromatic
number is about as large as the chromatic number, whereas if $t$ is greater then
it is close to the trivial upper bound $n/t$.
For $p\in (0,1)$ fixed, we obtain more precise information. In particular, we
find something more subtle happens at the threshold $t = \Theta(\log n)$, and we
determine the asymptotic first-order behaviour. Moreover, we consider the *$t$-component stability number*, the maximum order of a vertex subset that induces
a subgraph with maximum component order at most $t$, and show that it is
concentrated in a constant length interval about an explicitly given formula,
so long as $t = O(\log \log n)$.