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Inés Armendáriz

Publications and source records attributed to Inés Armendáriz.

13 recordsLinked to original sources
arXiv2026

Slot decomposition of continuous Box-Ball Systems

We study a piecewise constant function $η:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $ξ:\mathbb R\to\mathbb R$ satisfying $ξ'(x)=η(x)$, pinned by $ξ(0)=0$, 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 $ξ$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $ξ$ 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 $ξ$ 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↗
arXiv2023

Fluid limit for the coarsening phase of the condensing zero-range process

We prove a fluid limit for the coarsening phase of the condensing zero-range process on a finite number of sites. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries.

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↗
arXiv2021

Zero-range processes with rapidly growing rates

We provide two methods to construct zero-range processes with superlinear rates on ${\mathbb Z}^d$. In the first method these rates can grow very fast, if either the dynamics and the initial distribution are translation invariant or if only nearest neigbour translation invariant jumps are permitted, in the one-dimensional lattice. In the second method the rates cannot grow as fast but more general dynamics are allowed.

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↗
arXiv2016

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density in [J. Beltran, C. Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.

math.PR↗
arXiv2015

Phase transition for the dilute clock model

We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour $q$-state clock model in $\mathbb{Z}^d$, for every $q\geq 2$ and $d\geq 2$. This follows from the fact that the Edwards-Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma which is one of the main points of this article.

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↗
arXiv2011

Zero Range Condensation at Criticality

Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding or subtracting a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value at a critical scale. Our results also include distributional limits for the fluctuations of the maximum, which change from standard extreme value statistics to Gaussian when the density crosses the critical point. Fluctuations in the bulk are also covered, showing that the mass outside the maximum is distributed homogeneously. In summary, we identify the detailed behaviour at the critical scale including sub-leading terms, which provides a full understanding of the crossover from sub- to supercritical behaviour.

math.PR↗
arXiv2010

Brownian coagulation and a version of Smoluchowski's equation on the circle

We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion coefficients, the coagulation rates and the initial distribution of particles, we derive a spatially inhomogeneous version of the mass flow equation as the particle number tends to infinity. The mass flow equation is in one-to-one correspondence with Smoluchowski's coagulation equation. We prove uniqueness for this equation in a broad class of solutions, to which the weak limit of the stochastic system is shown to belong.

math.PR↗
arXiv2008

Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes

We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.

math.PR↗