arXiv subjects

Roberto Cortez

Publications and source records attributed to Roberto Cortez.

14 recordsLinked to original sources

arXiv2026

Relocation without preference: A destination-agnostic Schelling-type metapopulation model

In this work, we propose and analyze a novel Schelling-type metapopulation model that examines how random relocations of families between neighborhoods can lead to segregation. The model consists of a large number of houses organized into $N$ neighborhoods with $L$ houses each, without any spatial structure. Houses can be occupied by either a blue or a red family, and families relocate -- to an empty house selected uniformly at random -- at a rate that depends only on the number of families of the other type within the same neighborhood. We study two mean-field regimes: the large $N$ limit with fixed $L$, and the large $L$ limit with fixed $N$. The associated mean-field systems of ODEs are derived, and their long-time behavior is investigated. As is often the case with Schelling-type models, we find a rich interplay between the model parameters and the social structure of the equilibrium distribution, which exhibits segregation in some parameter ranges. Our work demonstrates that segregation patterns can emerge even when the relocation mechanism is destination-agnostic.

physics.soc-ph↗

arXiv2026

The fractal geometry of opinion formation

In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [9], within the framework of an interacting multi-agent system, where agents are assumed to interact with each other and update their opinions after each pairwise encounter. Specifically, our opinion model involves a large crowd of $N$ indistinguishable agents, each characterized by an opinion value ranging within the interval $[-1,1]$. At each update time, two agents are picked uniformly at random and the opinion of one agent will either shift by a proportion $\mu \in (0,1]$ towards $+1$, or by a proportion $\lambda \in (0,1]$ towards $-1$, with probabilities depending on the other agent's opinion. We rigorously derive the mean-field limit PDE that governs the large-population limit of the agent-based model and present several quantitative results demonstrating convergence to the unique equilibrium distribution. Remarkably, for a suitable choice of model parameters, the long-term equilibrium opinion profile displays a striking self-similar structure that generalizes the celebrated Bernoulli convolution, a topic extensively studied in the context of fractal geometry [24,51]. These findings also enhance our understanding of the opinion fragmentation phenomenon and may provide valuable insights for the development of more sophisticated models in future research.

math.PR↗

arXiv2024

Convergence and stationary distribution of Elo rating systems

The Elo rating system is a popular and widely adopted method for measuring the relative skill levels of players or teams in various sports and competitions. It assigns players numerical ratings and dynamically updates them based on game results and a model parameter $K$, which determines the sensitivity of rating changes. Assuming random games, this leads to a Markov chain for the evolution of the ratings of the $N$ players in the league. Despite its widespread use, little is known about the long-term behavior of this process. Aiming to fill this gap, in this article we prove that the process converges to its unique equilibrium distribution at an exponential rate in the 2-Wasserstein distance and almost surely. Moreover, we show important properties of the stationary distribution, such as the finiteness of an exponential moment, full support, and convergence to the players' true skills as $K$ decreases, at a rate of $\sqrt{K}$. We also provide Monte Carlo simulations that illustrate some of these properties and offer new insights.

math.PR↗

arXiv2024

Fractal opinions among interacting agents

We investigate an opinion model consisting of a large group of interacting agents, whose opinions are represented as numbers in $[-1,1]$. At each update time, two random agents are selected, and the opinion of the first agent is updated based on the opinion of the second (the ``persuader''). We derive the mean-field kinetic equation describing the large population limit of the model, and we provide several quantitative results establishing convergence to the unique equilibrium distribution. Surprisingly, in some range of the model parameters, the support of the equilibrium distribution exhibits a fractal structure. This provides a new mathematical description for the so-called opinion fragmentation phenomenon.

math.PR↗

arXiv2022

Uniform propagation of chaos for a dollar exchange econophysics model

We study the poor-biased model for money exchange introduced in [2]: agents are being randomly picked at a rate proportional to their current wealth, and then the selected agent gives a dollar to another agent picked uniformly at random. Simulations of a stochastic system of finitely many agents as well as a rigorous analysis carried out in [2,16] suggest that, when both the number of agents and time become large enough, the distribution of money among the agents converges to a Poisson distribution. In this manuscript, we establish a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which justifies the validity of the mean-field deterministic infinite system of ordinary differential equations as an approximation of the underlying stochastic agent-based dynamics.

math.PR↗

arXiv2022

Chaos for rescaled measures on Kac's sphere

In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac's sphere, which are obtained as the law of a vector of $N$ i.i.d. variables after it is rescaled to have unit average energy. We show that, as $N$ increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in $N$. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order $6$ to $4+ε$.

math.PR↗

arXiv2022

SIR model with social gatherings

We introduce an extension to Kermack and McKendrick's classic susceptible-infected-recovered (SIR) model in epidemiology, whose underlying mechanism of infection consists of individuals attending randomly generated social gatherings. This gives rise to a system of ODEs where the force of infection term depends non-linearly on the proportion of infected individuals. Some specific instances yield models already studied in the literature, to which the present work provides a probabilistic foundation. The basic reproduction number is seen to depend quadratically on the average size of the gatherings, which may be helpful to understand how restrictions on social gatherings affect the spread of the disease. We rigorously justify our model by showing that the system of ODEs is the mean-field limit of the jump Markov process corresponding to the evolution of the disease in a finite population.

math.PR↗

arXiv2021

Persistence in a large network of locally interacting neurons

This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of local interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear meanfield process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.

math.PR↗

arXiv2020

On a thermostated Kac model with rescaling

We introduce a global thermostat on Kac's 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation. We prove ergodicity for this equation, and show that its equilibrium distribution has a density that, depending on the parameters of the model, can exhibit heavy tails, and whose behaviour at the origin can range from being analytic, to being $C^k$, and even to blowing-up. Finally, we prove propagation of chaos for the associated $N$-particle system, with a uniform-in-time rate of order $N^{-η}$ in the squared $2$-Wasserstein metric, for an explicit $η\in (0, 1/3]$.

math-ph↗

arXiv2020

Uniform propagation of chaos for the thermostated Kac model

We consider Kac's 1D N-particle system coupled to an ideal thermostat at temperature T, introduced by Bonetto, Loss, and Vaidyanathan in 2014. We obtain a propagation of chaos result for this system, with explicit and uniform-in-time rates of order N^(-1/3) in the 2-Wasserstein metric. We also show well-posedness and equilibration for the limit kinetic equation in the space of probability measures. The proofs use a coupling argument previously introduced by Cortez and Fontbona in 2016.

math.PR↗

arXiv2018

Particle system approach to wealth redistribution

We study a stochastic $N$-particle system representing economic agents in a population randomly exchanging their money, which is associated to a class of one-dimensional kinetic equations modelling the evolution of the distribution of wealth in a simple market economy, introduced by Matthes and Toscani \cite{matthes-toscani2008}. We show that, unless the economic exchanges satisfy some exact conservation condition, the $p$-moments of the particles diverge with time for all $p>1$, and converge to 0 for $0<p<1$. This establishes a qualitative difference with the kinetic equation, whose solution is known to have bounded $p$-moments, for all $p$ smaller than the Pareto index of the equilibrium distribution. On the other hand, the case of strictly conservative economies is fully treated: using probabilistic coupling techniques, we obtain stability results for the particle system, such as propagation of moments, exponential equilibration, and uniform (in time) propagation of chaos with explicit rate of order $N^{-1/3}$.

math.PR↗

arXiv2017

Quantitative uniform propagation of chaos for Maxwell molecules

We prove propagation of chaos at explicit polynomial rates in Wasserstein distance W_2 for Kac's N-particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules, with and without cutoff. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost N^{-1/3} for W_2^2.

math-ph↗

arXiv2016

Uniform propagation of chaos for Kac's 1D particle system

In this paper we study Kac's 1D particle system, consisting of the velocities of $N$ particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in $N$, for both the squared (i.e., the energy) and non-squared particle system. These rates are of order $N^{-1/3}$ (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order ($4+ε$ is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.

math.PR↗

arXiv2016

Quantitative propagation of chaos for generalized Kac particle systems

We study a class of one-dimensional particle systems with true (Bird type) binary interactions, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of nonindependent nonlinear processes, as well as on recent sharp estimates for empirical measures.

math.PR↗