arXiv subjects

Pablo A. Ferrari

Publications and source records attributed to Pablo A. Ferrari.

At least 19 recordsLinked to original sources

arXiv2026

Slot decomposition of continuous Box-Ball Systems

We study a piecewise constant function $\eta:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $\xi:\mathbb R\to\mathbb R$ satisfying $\xi'(x)=\eta(x)$, pinned by $\xi(0)=0$, has finite length excursions over past minima. This is the continuous generalization of an initial ball configuration in the discrete Box Ball System introduced by Takahashi and Satsuma, where solitons of integer sizes $k\ge1$ are identified. We extend the slot decomposition developed by Ferrari, Nguyen, Rolla and Wang in the discrete setting to the continuous case. Each soliton of $\xi$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $\xi$ to a point configuration. We consider a distribution on walks given by a product measure on the decomposition of the path into excursions over past minima. Excursions are distributed as products of their solitons weights, which are determined by the soliton heights. We show that when the weight function is in $L^1$ the slot decomposition of $\xi$ is a Poisson process. This extends to the continuous case an approach of Ferrari and Gabrielli. As an example, we compute the intensity measure of the Poisson process associated to the asymmetric telegraph process introduced by Kac. In a forthcoming paper we discuss the dynamic properties.

math.PR↗

arXiv2025

Gibbs conditioning principle for log-concave independent random variables

Let $\nu_1,\nu_2,\dots$ be a sequence of probabilities on the nonnegative integers, and $X=(X_1,X_2, \dots)$ be a sequence of independent random variables $X_i$ with law $\nu_i$. For $\lambda>0$ denote $Z^\lambda_i:= \sum_x \lambda^x\nu_i(x)$ and $\lambda^{\max}:= \sup\{\lambda>0: Z^\lambda_i<\infty \text{ for all }i\}$, and assume $\lambda^{\max}>1$. For $\lambda<\lambda^{\max}$, define the tilted probability $\nu_i^{\lambda}(x):= \lambda^x\nu_i(x)/Z^{\lambda}_i$, and let $X^\lambda$ be a sequence of independent variables $X^\lambda_i$ with law $\nu^{\lambda}_i$, and denote $S^\lambda_n:= X^{\lambda}_1+\dots+X^{\lambda}_n$, with $S_n=S^1_n$. Choose $\lambda^*\in(1,\lambda^{\max})$ and denote $R^*_n:= E (S^{\lambda^*}_n)$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in\cdot|S_n>R^*_n)$ converges weakly to the law of $X^{\lambda^*}$, as $n\to\infty$. We prove the GCP for log-concave $\nu_i$'s, meaning $\nu_i(x+1)\,\nu_i(x-1) \le ( \nu_i(x))^2$, subject to a technical condition that prevents condensation. The canonical measures are the distributions of the first $n$ variables, conditioned on their sum being $k$. Efron's theorem states that for log-concave $\nu_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^\lambda\in\cdot|S^\lambda_n>R^*_n)$ in terms of $\lambda$. This ordering is a fundamental component of our proof.

math.PR↗

arXiv2025

Multitime fields and hard rod scaling limits

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 with each line of the process, a random surface called multitime walk field is obtained. The diffusive rescaling of the surface converges to the multitime Brownian motion, a classical Gaussian field also called L\'evy-Chentsov field. A cut of the multitime fields with a perpendicular plane, reveals a one dimensional continuous time random walk and a Brownian motion, respectively. A hard rod is an interval contained in $\mathbb{R}$ that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes. When rod sizes are zero we have an ideal gas dynamics. We describe the relation between ideal gas and hard-rod invariant measures.

math.PR↗

arXiv2024

Hidden temperature in the KMP model

In the Kipnis Marchioro Presutti (KMP) model a positive energy $ζ_i$ is associated with each vertex $i$ of a finite graph with a boundary. When a Poisson clock rings at an edge $ij$ with energies $ζ_i,ζ_j$, those values are substituted by $U(ζ_i+ζ_j)$ and $(1-U)(ζ_i+ζ_j)$, respectively, where $U$ is a uniform random variable in $(0,1)$. A value $T_j\ge 0$ is fixed at each boundary vertex $j$. The dynamics is defined in such way that the resulting Markov process $ζ(t)$, satisfies that $ζ_j(t)$ is exponential with mean $T_j$, for each boundary vertex $j$, for all $t$. We show that the invariant measure is the distribution of a vector $ζ$ with coordinates $ζ_i=T_i X_i$, where $X_i$ are iid exponential$(1)$ random variables, the law of $T$ is the invariant measure for an opinion random averaging/gossip model with the same boundary conditions of $ζ$, and the vectors $X$ and $T$ are independent. The result confirms a conjecture based on the large deviations of the model. When the graph is one-dimensional, we bound the correlations of the invariant measure and perform the hydrostatic limit. We show that the empirical measure of a configuration chosen with the invariant measure converges to the linear interpolation of the boundary values.

math.PR↗

arXiv2023

Macroscopic diffusive fluctuations for generalized hard rods dynamics

We study the fluctuations in equilibrium for a dynamics of rods with random length. This includes the classical hard rod elastic collisions, when rod lengths are constant and equal to a positive value. We prove that in the diffusive space-time scaling, an initial fluctuation of density of particles of velocity $v$, after recentering on its Euler evolution, evolve randomly shifted by a Brownian motion of variance $\mathcal D(v)$.

math-ph↗

arXiv2022

Hard Rod Hydrodynamics and the Levy Chentsov Field

We study the hydrodynamics of the hard rod model proposed by Boldrighini, Dobrushin and Soukhov by describing the displacement of each quasiparticle with respect to the corresponding ideal gas particle as a height difference in a related field. Starting with a family of nonhomogeneous Poisson processes contained in the position-velocity-length space $\mathbb{R}^3$, we show laws of large numbers for the quasiparticle positions and the length fields, and the joint convergence of the quasiparticle fluctuations to a Levy Chentsov field. We allow variable rod lengths, including negative lengths.

math.PR↗

arXiv2021

Group testing with nested pools

In order to identify the infected individuals of a population, their samples are divided in equally sized groups called pools and a single laboratory test is applied to each pool. Individuals whose samples belong to pools that test negative are declared healthy, while each pool that tests positive is divided into smaller, equally sized pools which are tested in the next stage. In the $(k+1)$-th stage all remaining samples are tested. If $p<1-3^{-1/3}$, we minimize the expected number of tests per individual as a function of the number $k+1$ of stages, and of the pool sizes in the first $k$ stages. We show that for each $p\in (0, 1-3^{-1/3})$ the optimal choice is one of four possible schemes, which are explicitly described. We conjecture that for each $p$, the optimal choice is one of the two sequences of pool sizes $(3^k\text{ or }3^{k-1}4,3^{k-1},\dots,3^2,3 )$, with a precise description of the range of $p$'s where each is optimal. The conjecture is supported by overwhelming numerical evidence for $p>2^{-51}$. We also show that the cost of the best among the schemes $(3^k,\dots,3)$ is of order $O\big(p\log(1/p)\big)$, comparable to the information theoretical lower bound $p\log_2(1/p)+(1-p)\log_2(1/(1-p))$, the entropy of a Bernoulli$(p)$ random variable.

math.ST↗

arXiv2021

Gaussian random permutation and the boson point process

We construct an infinite volume spatial random permutation $(\mathsf X,σ)$, where $\mathsf X\subset\mathbb R^d$ is locally finite and $σ:\mathsf X\to \mathsf X$ is a permutation, associated to the formal Hamiltonian $$ H(\mathsf X,σ) = \sum_{x\in \mathsf X} \|x-σ(x)\|^2. $$ The measures are parametrized by the point density $ρ$ and the temperature $α$. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Let $ρ_c=ρ_c(α)$ be the critical density for Bose-Einstein condensation in Feynman's representation. Each finite cycle of $σ$ induces a loop of points of~$\mathsf X$. For $ρ\le ρ_c$ we define $(\mathsf X, σ)$ as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For $d\ge 3$ and $ρ>ρ_c$ we define $(\mathsf X,σ)$ as the superposition of independent realizations of the Gaussian loop soup at density $ρ_c$ and the Gaussian random interlacements at density $ρ-ρ_c$. In either case we call $(\mathsf X, σ)$ a Gaussian random permutation at density $ρ$ and temperature $α$. The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian $H$. Its point marginal $\mathsf X$ has the same distribution as the boson point process introduced by Shirai-Takahashi (2003) in the subcritical case, and by Tamura-Ito (2007) in the supercritical case.

math-ph↗

arXiv2020

Slow-to-Start Traffic Model: Condensation, Saturation and Scaling Limits

We consider a one-dimensional traffic model with a slow-to-start rule. The initial position of the cars in $\mathbb R$ is a Poisson process of parameter $λ$. Cars have speed 0 or 1 and travel in the same direction. At time zero the speed of all cars is 0; each car waits an exponential time to switch speed from $0$ to $1$ and stops when it collides with a stopped car. When the car is no longer blocked, it waits a new exponential time to assume speed one, and so on. We study the emergence of condensation for the saturated regime $λ>1$ and the critical regime $λ=1$, showing that in both regimes all cars collide infinitely often and each car has asymptotic mean velocity $1/λ$. In the saturated regime the moving cars form a point process whose intensity tends to 1. The remaining cars condensate in a set of points whose intensity tends to zero as $1/\sqrt t$. We study the scaling limit of the traffic jam evolution in terms of a collection of coalescing Brownian motions.

math.PR↗

arXiv2019

BBS invariant measures with independent soliton components

The Box-Ball System (BBS) is a one-dimensional cellular automaton in $\{0,1\}^\Z$ introduced by Takahashi and Satsuma \cite{TS}, who also identified conserved sequences called \emph{solitons}. Integers are called boxes and a ball configuration indicates the boxes occupied by balls. For each integer $k\ge1$, a $k$-soliton consists of $k$ boxes occupied by balls and $k$ empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang \cite{FNRW} define the $k$-slots of a configuration as the places where $k$-solitons can be inserted. Labeling the $k$-slots with integer numbers, they define the $k$-component of a configuration as the array $\{ζ_k(j)\}_{j\in \mathbb Z}$ of elements of $\Z_{\ge0}$ giving the number $ζ_k(j)$ of $k$-solitons appended to $k$-slot $j\in \mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $μ$ has independent soliton components, then $μ$ is invariant for the automaton. We show that for each $λ\in[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $λ$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(λ)$, an explicit function of $λ$. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.

math.PR↗

arXiv2019

Gibbs measures over permutations of point processes with low density

We study a model of spatial random permutations over a discrete set of points. Formally, a permutation $σ$ is sampled proportionally to the weight $\exp\{-α\sum_x V(σ(x)-x)\},$ where $α>0$ is the temperature and $V$ is a non-negative and continuous potential. The most relevant case for physics is when $V(x)=\|x\|^2$, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in $\mathbb{Z}^d$ with intensity $ρ\in(0,1/2)$, and the potential verifies some regularity conditions. In particular, we prove that if $α$ is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in $\mathbb{R}^d$ with low enough intensity.

math.PR↗

arXiv2018

Yaglom Limit via Holley Inequality

Let $S$ be a countable set provided with a partial order and a minimal element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a quasi-stationary distribution. We use Holley inequality to obtain sufficient conditions under which the following hold. The trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on $S$, when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.

math.PR↗

arXiv2018

Soliton decomposition of the Box-Ball System

The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size $k$ solitons in each $k$-slot. The dynamics of the components is linear: the $k$-th component moves rigidly at speed $k$. Let $\zeta$ be a translation invariant family of independent random vectors under a summability condition and $\eta$ the ball configuration with components $\zeta$. We show that the law of $\eta$ is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac12$. We also show that starting BBS with an ergodic measure, the position of a tagged $k$-soliton at time $t$, divided by $t$ converges as $t\to\infty$ to an effective speed $v_k$. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.

math-ph↗

arXiv2018

TASEP hydrodynamics using microscopic characteristics

The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.

math.PR↗

arXiv2017

Hydrodynamics of the $N$-BBM process

The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in $\mathbb R$ and each particle at rate 1 creates a new particle at her current position; the newborn particle increments and branchings are independent of the other particles. The $N$-BBM starts with $N$ particles and at each branching time, the leftmost particle is removed so that the total number of particles is $N$ for all times. The $N$-BBM was proposed by Maillard and belongs to a family of processes introduced by Brunet and Derrida. We fix a density $ρ$ with a left boundary $L=\sup\{r\in\mathbb R: \int_r^\infty ρ(x)dx=1\}>-\infty$ and let the initial particle positions be iid continuous random variables with density $ρ$. We show that the empirical measure associated to the particle positions at a fixed time $t$ converges to an absolutely continuous measure with density $ψ(\cdot,t)$, as $N\to\infty$. The limit $ψ$ is solution of a free boundary problem (FBP) when this solution exists. The existence of solutions for finite time-intervals has been recently proved by Lee.

math.PR↗

arXiv2016

Perfect Necklaces

We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under rotations. For positive integers k and n, we call a necklace (k,n)-perfect if each word of length k occurs exactly n times at positions which are different modulo n for any convention on the starting point. We call a necklace perfect if it is (k,k)-perfect for some k. We prove that every arithmetic sequence with difference coprime with the alphabet size induces a perfect necklace. In particular, the concatenation of all words of the same length in lexicographic order yields a perfect necklace. For each k and n, we give a closed formula for the number of (k,n)-perfect necklaces. Finally, we prove that every infinite periodic sequence whose period coincides with some (k,n)-perfect necklace for any n, passes all statistical tests of size up to k, but not all larger tests. This last theorem motivated this work.

math.CO↗

arXiv2014

Separation versus diffusion in a two species system

We consider a finite number of particles that move in $\mathbb Z$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$ particle becomes a $b$ particle at constant rate, while the leftmost $b$ particle becomes $a$ particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a non linear system of two PDE's with free boundaries.

math.PR↗

arXiv2014

Finite cycle Gibbs measures on permutations of $\mathbb Z^d$

We consider Gibbs distributions on the set of permutations of $\mathbb Z^d$ associated to the Hamiltonian $H(σ):=\sum_{x} V(σ(x)-x)$, where $σ$ is a permutation and $V:\mathbb Z^d\to\mathbb R$ is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on $V$ ensuring that for large enough temperature $α>0$ there exists a unique infinite volume ergodic Gibbs measure $μ^α$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $μ^α$ as the unique invariant measure of a Markov process on the set of finite-cycle permutations that can be seen as a loss-network, a continuous-time birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define $τ_v$ as the shift permutation $τ_v(x)=x+v$. In the Gaussian case $V=\|\cdot\|^2$, we show that for each $v\in\mathbb Z^d$, $μ^α_v$ given by $μ^α_v(f)=μ^α[f(τ_v\cdot)]$ is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $τ_v$ boundary conditions. For a general potential $V$, we prove the existence of Gibbs measures $μ^α_v$ when $α$ is bigger than some $v$-dependent value.

math.PR↗