arXiv subjects

Josué Corujo

Publications and source records attributed to Josué Corujo.

8 recordsLinked to original sources

arXiv2026

A novel approach to the giant component fluctuations

We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walk, introduced by Limic in 2019, that encodes the dynamic of the evolution of the sizes of the connected components of a large class of random graph processes. We limit our study to the variant of the Erdős-Rényi graph process $(G_n(s))_{s\geq 0}$ with $n$ vertices where an edge connecting a pair of vertices appears at an exponential rate 1 waiting time, independently over pairs. We first use the properties of the simultaneous breadth-first walk to obtain an alternative and self-contained proof of the functional central limit theorem recently established by Enriquez, Faraud and Lemaire in the super-critical regime ($s=\frac{c}{n}$ and $c>1$). Next, to show the versatility of our approach, we prove a functional central limit theorem in the barely super-critical regime ($s=\frac{1+tε_n}{n}$ where $t>0$ and $(ε_n)_n$ is a sequence of positive reals that converges to 0 such that $(nε_n^3)_n$ tends to $+\infty$).

math.PR↗

arXiv2026

A dynamical approach to spanning and surplus edges of random graphs

Consider a finite inhomogeneous random graph evolving in continuous time, where each vertex is assigned a mass, and an edge between any pair of vertices appears at a rate proportional to the product of their masses. The process tracking the evolution of component sizes evolves according to the multiplicative coalescent dynamic and can be encoded using the simultaneous breadth-first walk introduced by Limic (2019). We extend this encoding to incorporate surplus edge data within each connected component. Two distinct graph-based representations of the multiplicative coalescent, each with its own advantages and limitations, are analyzed in detail. In particular, a canonical multigraph introduced by Bhamidi, Budhiraja and Wang (2014), which is naturally connected to the augmented multiplicative coalescent, emerges from our framework. We demonstrate that a transformation of the simultaneous breadth-first walk, supplemented with an additional and independent source of randomness, encodes the full dynamics of the augmented multiplicative coalescent.

math.PR↗

arXiv2026

The number of connected components in sub-critical random graph processes

We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is applied to several examples. In the particular setting of the Erdős-Rényi graph process, we explicit the fluid limit of the number of connected components normalized, and the diffusion limit of the scaled fluctuations in the sub-critical regime, where the mean degree is between zero and one.

math.PR↗

arXiv2025

Matching random colored points with rectangles (Corrigendum)

Given $n>0$, let $S\subset [0,1]^2$ be a set of $n$ points, chosen uniformly at random. Let $R\cup B$ be a random partition, or coloring, of $S$ in which each point of $S$ is included in $R$ uniformly at random with probability $1/2$. Corujo et al.~(JOCO 2023) studied the random variable $M(n)$ equal to the number of points of $S$ that are covered by the rectangles of a maximum matching of $S$ using pairwise-disjoint rectangles. Each rectangle is axis-aligned and covers exactly two points of $S$ of the same color. They designed a deterministic algorithm to match points of $S$, and the algorithm was modeled as a discrete stochastic process over a finite set of states. After claiming that the stochastic process is a Markov chain, they proved that almost surely $M(n)\ge 0.83\,n$ for $n$ large enough. The issue is that such a process is not actually a Markov one, as we discuss in this note. We argue this issue, and correct it by obtaining the same result but considering that the stochastic process is not Markov, but satisfies some kind of first-order homogeneity property that allows us to compute its marginal distributions.

cs.CG↗

arXiv2023

The standard augmented multiplicative coalescent revisited

The Erdős-Rényi random graph is the fundamental random graph model. In this paper we consider its continuous-time version, where multi-edges and self-loops are also allowed. It is well-known that the sizes of its connected components evolve according to the multiplicative coalescent dynamics. Moreover, with the additional information on the number of surplus edges, the resulting process follows the augmented multiplicative coalescent dynamic, constructed by Bhamidi, Budhiraja and Wang in 2014. The same authors exhibit the scaling limit, which can be specified in terms of the infinite vector of excursions (in particular their lengths, and the areas enclosed by the excursion curves) above past infima of a reflected Brownian motion with linear infinitesimal drift. We use some recent results, using a graph exploration process called the simultaneous breadth-first walk, to study the same scaling limit, called the standard augmented multiplicative coalescent (SAMC). We present a self-contained, simpler and more direct approach than that of any previous construction of the SAMC. Furthermore, we believe that the method described here is convenient for generalizations, one of which would be the study of general non-standard eternal augmented multiplicative coalescents.

math.PR↗

arXiv2023

On the spectrum and ergodicity of a neutral multi-allelic Moran model

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.

math.PR↗

arXiv2022

Uniform in time propagation of chaos for a Moran model

The goal of this article is to study the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. The rate matrix driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided into two parts. The first one deals with processes with general selection rates. For this case we are able to prove the propagation of chaos in $\mathbb{L}^p$ over the compacts, with speed of convergence of order $1/\sqrt{N}$. Further on, we consider a specific type of selection that we call additive selection. Essentially, we assume that the selection rate can be decomposed as the sum of three terms: a term depending on the allelic type of the parent (which can be understood as selection at death), another term depending on the allelic type of the descendant (which can be understood as selection at birth) and a third term which is symmetric. Under this setting, our results include a uniform in time bound for the propagation on chaos in $\mathbb{L}^p$ of order $1/\sqrt{N}$, and the proof of the asymptotic normality with zero mean and explicit variance, for the approximation error between the empirical distribution and its limit, when the number of individuals tend towards infinity. Additionally, we explore the interpretation of the Moran model with additive selection as a particle process whose empirical distribution approximates a quasi-stationary distribution, in the same spirit as the Fleming\,--\,Viot particle systems. We then address the problem of minimising the asymptotic quadratic error, when the time and the number of particles go to infinity.

math.PR↗

arXiv2021

Dynamics of a Fleming-Viot type particle system on the cycle graph

We study the Fleming-Viot particle process formed by N interacting continuous-time asymmetric random walks on the cycle graph, with uniform killing. We show that this model has a remarkable exact solvability, despite the fact that it is non-reversible with non-explicit invariant distribution. Our main results include quantitative propagation of chaos and exponential ergodicity with explicit constants, as well as formulas for covariances at equilibrium in terms of the Chebyshev polynomials. We also obtain a bound uniform in time for the convergence of the proportion of particles in each state when the number of particles goes to infinity.

math.PR↗