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Nathanaël Enriquez

Publications and source records attributed to Nathanaël Enriquez.

At least 19 recordsLinked to original sources
arXiv2026

Optimal matching under size priority

Past studies on the local limit of maximal weight matchings in edge-weighted large random graphs rely fundamentally on the assumption that the weights are atomless, which ensures that the maximal weight matching is unique. This excludes de facto maximal size matchings that correspond to equal edge-weights. In this work, we overcome this difficulty by assigning i.i.d.~atomless weights to edges and choosing the maximal size matching that maximises the weight. We call these doubly constrained matchings \emph{optimal matchings}. The natural generalisation of optimal matchings for infinite unimodular random graphs are unimodular matchings of maximal density at the root that maximise the expected weight at the root when it is matched. For unimodular Bienaymé-Galton-Watson (UBGW) trees and for a broad class of weight distributions, we show existence and uniqueness in law of such matchings. We also prove that if a sequence of finite random weighted graphs converges locally to an UBGW tree with i.i.d.~weights, then there exists a sequence of matchings on the finite graphs that converges locally to the optimal matching on the limiting tree. Finally, we identify a regime, depending only on the offspring distribution of the limiting tree, in which correlations between edge states in the optimal matching decay exponentially with their graph distance. In this regime, we strengthen the previous convergence to the convergence of optimal matchings of the finite graphs. As a by-product, we can explicitly compute the asymptotic densities of edges that belong to all maximal-density matchings, and of edges that belong to none.

math.PR↗
arXiv2025

Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

We consider the typical Poisson-Voronoi cell in the Euclidean space R d and in particular the maximal distance D from a vertex of that cell to its nucleus. We provide a sharp asymptotics for the tail distribution of D. As a byproduct, we prove that the extremal index related to the sequence of such distances for all Voronoi cells included in a large box is equal to (2d) -1 . This confirms a conjecture formulated by Chenavier and Robert. The explicit constant appearing in the estimate of the tail probability of D is proved to be the mean volume of a random simplex formed by uniformly distributed points on the unit sphere conditioned on satisfying some spatial condition.

math.PR↗
arXiv2025

Ideal Poisson-Voronoi tessellations on hyperbolic spaces

We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces $ \mathbb{H}_{d}$ for $d \geq 2$. In contrast to the Euclidean setting, a limiting nontrivial ideal tessellation $ \mathcal{V}_{d}$ appears as the intensity tends to $0$. The tessellation $ \mathcal{V}_{d}$ is a natural, isometry-invariant decomposition of $ \mathbb{H}_{d}$ into countably many unbounded polytopes, each with a unique end. We study its basic properties, in particular, the geometric features of its cells.

math.PR↗
arXiv2025

Optimal Unimodular Matching

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.

math.PR↗
arXiv2024

The process of fluctuations of the giant component of an Erdős-Rényi graph

We present a detailed study of the evolution of the giant component of the Erdős-Rényi graph process as the mean degree increases from 1 to infinity. It leads to the identification of the limiting process of the rescaled fluctuations of its order around its deterministic asymptotic. This process is Gaussian with an explicit covariance.

math.PR↗
arXiv2022

Reversible Poisson-Kirchhoff Systems

We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.

math.PR↗
arXiv2022

Depth First Exploration of a Configuration Model

We introduce an algorithm that constructs a random uniform graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.

math.PR↗
arXiv2022

Elongated Poisson-Voronoi cells in an empty half-plane

The Voronoi tessellation of a homogeneous Poisson point process in the lower half-plane gives rise to a family of vertical elongated cells in the upper half-plane. The set of edges of these cells is ruled by a Markovian branching mechanism which is asymptotically described by two sequences of iid variables which are respectively Beta and exponentially distributed. This leads to a precise description of the scaling limit of a so-called typical cell. The limit object is a random apeirogon that we name menhir in reference to the Gallic huge stones. We also deduce from the aforementioned branching mechanism that the number of vertices of a cell of height $λ$ is asymptotically equal to $\frac45\logλ$.

math.PR↗
arXiv2021

Long induced paths in a configuration model

In an article published in 1987 in Combinatorica \cite{MR918397}, Frieze and Jackson established a lower bound on the length of the longest induced path (and cycle) in a sparse random graph. Their bound is obtained through a rough analysis of a greedy algorithm. In the present work, we provide a sharp asymptotic for the length of the induced path constructed by their algorithm. To this end, we introduce an alternative algorithm that builds the same induced path and whose analysis falls into the framework of a previous work by the authors on depth-first exploration of a configuration model \cite{EFMN}. We also analyze an extension of our algorithm that mixes depth-first and breadth-first explorations and generates $m$-induced paths.

math.PR↗
arXiv2020

A solvable class of renewal processes

When the distribution of the inter-arrival times of a renewal process is a mixture of geometric laws, we prove that the renewal function of the process is given by the moments of a probability measure which is explicitly related to the mixture distribution. We also present an analogous result in the continuous case when the inter-arrival law is a mixture of exponential laws. We then observe that the above discrete class of renewal processes provides a solvable family of random polymers. Namely, we obtain an exact representation of the partition function of polymers pinned at sites of the aforementioned renewal processes. In the particular case where the mixture measure is a generalized Arcsine law, the computations can be explicitly handled.

math.PR↗
arXiv2020

Eigenvectors of a matrix under random perturbation

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to the state defined by a given vector.

math.PR↗
arXiv2018

Mean asymptotics for a Poisson-Voronoi cell on a Riemannian manifold

In this paper, we consider a Riemannian manifold $M$ and the Poisson-Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $λ$ on $M$. We obtain asymptotic expansions up to the second order for the means of several characteristics of the Voronoi cell associated with $x_0$, including its volume and number of vertices. In each case, the first term of the estimate is equal to the mean characteristic in the Euclidean setting while the second term may contain a particular curvature of $M$ at $x_0$: the scalar curvature in the case of the mean number of vertices, the Ricci curvature in the case of the density of vertices and the sectional curvatures in the cases of the volume and number of vertices of a section of the Voronoi cell. Several explicit formulas are also derived in the particular case of constant curvature. The key tool for proving these results is a new change of variables formula of Blaschke-Petkantschin type in the Riemannian setting. Finally, a probabilistic proof of the Gauss-Bonnet Theorem is deduced from the asymptotic estimate of the total number of vertices of the tessellation in dimension two.

math.PR↗
arXiv2017

Empirical spectral distribution of a matrix under perturbation

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are either related to the one-dimensional Gaussian free field or to free probability theory.

math.PR↗
arXiv2017

Limiting shape of the Depth First Search tree in an Erdős-Rényi graph

We show that the profile of the tree constructed by the Depth First Search Algorithm in the giant component of an Erdős-Rényi graph with $N$ vertices and connection probability $c/N$ converges to an explicit deterministic shape. This makes it possible to exhibit a long non-intersecting path of length $\left( ρ_c - \frac{\mathrm{Li}_2(ρ_c)}{c} \right) \times N$, where $ρ_c$ is the density of the giant component.

math.PR↗
arXiv2016

The probability that two random integers are coprime

An equivalence is proven between the Riemann Hypothesis and the speed of convergence to 1/zeta(2) of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0.

math.PR↗
arXiv2016

On the number of lattice convex chains

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeros of the zeta function and leads to a necessary and sufficient condition for the Riemann Hypothesis to hold.

math.PR↗
arXiv2016

The Poisson-Voronoi cell around an isolated nucleus

Consider a planar random point process made of the union of a point (the origin) and of a Poisson point process with a uniform intensity outside a deterministic set surrounding the origin. When the intensity goes to infinity, we show that the Voronoi cell associated with the origin converges from above to a deterministic convex set. We describe this set and give the asymptotics of the expectation of its defect area, defect perimeter and number of vertices. On the way, two intermediary questions are treated. First, we describe the mean characteristics of the Poisson-Voronoi cell conditioned on containing a fixed convex body around the origin and secondly, we show that the nucleus of such cell converges to the Steiner point of the convex body. As in Rényi and Sulanke's seminal papers on random convex hulls, the regularity of the convex body has crucial importance. We deal with both the smooth and polygonal cases. Techniques are based notably on accurate estimates of the area of the Voronoi flower and of the support function of the cell containing the origin as well as on an Efron-type relation.

math.PR↗