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The alpha-stable component

With Bénédicte Haas et Christina Goldschmidt, we studied the properties of a random metric space that we call the alpha-stable component. This metric space appears as the scaling limit of a large connected component of a critical configuration model with i.i.d degrees with a heavy tail, parametrised by a number alpha between 1 and 2. It is almost a tree up to a finite number of cycles, which we call the surplus.

Alpha=1.5, surplus=2

Alpha=1.7, surplus=2

Random gluig of metric spaces

In an ongoing work The images that follow have been generated using the model studied in my paper Random gluing of metric spaces, published in the Annals of Probability. Here the weight sequence (w_n) is chosen to be w_n=n^-3/2 for the three simulations. The radii (lambda_n) of the sequence of circles/spheres used in the two static pictures are taken to be n^-1/2. In the animated picture, we coupled the construction for different sequences of radii, namely lambda_n=n^-alpha, with alpha between 0.1 and 1.

Random trees constructed by aggregation

The trees below are constructed using the random construction described in the paper Random trees constructed by aggregation of Curien and Haas. We fix a sequence (a_i) of positive real numbers and the construction is described recursively. We start at time 1 with a segment of length a_1. At time n+1, we sample a uniform point on the tree constructed so far and we glue a segment of length a_{n+1} at the chosen location. The pictures depict the tree obtained after 1000 iterations of this procedure. For the picture on the left, we chose a_i=i^-1/2; for the one on the right we chose a_i=i^-1/3.

Minimal spanning tree of the complete graph

Here is a simulation of the minimal spanning tree of a complete graph with 30.000 vertices. According to this article of Addario-Berry, Broutin, Goldschmidt and Miermont, this random tree rescaled by n^1/3, where n is the number of vertices, converges in distribution to a continuum random tree. This simulation originates from an interesting discussion with Othmane Safsafi about the possible self-similarity in distribution of such trees, during Saint-Flour Summer School in 2017.

Large connected components of a critical configuration model

Here are some simulations that I ran during my research internship with Chistina Goldschmidt in 2015. These pictures depict large connected components taken from random graphs that were constructed using a configuration model with i.i.d. degrees, using a degree distribution which has a polynomial tail and is critical in some sense (see this article of Adrien Joseph which proves the convergence in distribution of the size (number of vertices) of those components. This model depends on a parameter gamma between 3 and 4.

Gamma = 3.5

Gamma = 3.8

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