Related papers: High-dimensional permutons: theory and application…
We study the upper and lower regularity dimensions in relation to the notions of doubling and uniformly perfect. These two regularity properties are closely related which is quantified thanks to the regularity dimensions. The regularity…
In this paper, we analyze $m$-dimensional ($m$D) convolutional codes with finite support, viewed as a natural generalization of one-dimensional (1D) convolutional codes to higher dimensions. An $m$D convolutional code with finite support…
We study two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation $\sigma$ is a pair of consecutive elements $\sigma(i), \sigma(i+1)$ such that $\sigma(i) >…
We consider Bienaym\'e-Galton-Watson trees in random environment, where each generation $k$ is attributed a random offspring distribution $\mu_k$, and $(\mu_k)_{k\geq 0}$ is a sequence of independent and identically distributed random…
In this paper we investigate the scaling limit of the range (the set of visited vertices) for a class of critical lattice models, starting from a single initial particle at the origin. We give conditions on the random sets and an associated…
We review a recent development at the interface between discrete mathematics on one hand and probability theory and statistics on the other, specifically the use of Markov chains and their boundary theory in connection with the asymptotics…
Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph, and use it to develop a continuum perturbation expansion for the MST on…
In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…
A Poisson line process is a random set of straight lines contained in the plane, as the image of the map $(x,v)\mapsto (x+vt)_{t\in\mathbb{R}}$, for each point $(x,v)$ of a Poisson process in the space-velocity plane. By associating a step…
We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite…
Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to $k$th…
Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering and life sciences. In this work, we investigate the statistical properties of…
Random systems of curves exhibiting fluctuating features on arbitrarily small scales ($\delta$) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply…
We study non-uniform percolation in a two-dimensional cluster growth model with multiple seeds. With increasing concentration of seeds, the percolation threshold is found to increase monotonically, while the exponents for correlation…
Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics,…
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…
Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $D\geq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$…
During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by {\it the same} universal probability distribution…