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.Comment: 14 pages, 2 figure

BBS invariant measures with independent soliton components

The Box-Ball System (BBS) is a one-dimensional cellular automaton in the configuration space { 0 , 1 } ^Z introduced by Takahashi and Satsuma [8], who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang [4] map a configuration to a family of soliton components, indexed by the soliton sizes k ≥ 1 . Building over this decomposition, we give an explicit construction of a large family of invariant measures for the BBS that are also shift invariant, including Ising-like Markov and Bernoulli product measures. The construction is based on the concatenation of iid excursions of the associated walk trajectory. Each excursion has the property that the law of its k component given the larger components is product of a finite number of geometric distributions with a parameter depending on k . As a consequence, the law of each component of the resulting ball configuration is product of identically distributed geometric random variables, and the components are independent. This last property implies invariance for BBS, as shown by [4].Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Gabrielli, Davide. Universita degli Studi dell'Aquila; Itali

Box-ball system: soliton and tree decomposition of excursions

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514–3519, 1990). Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari et al. (Soliton decomposition of the box-ball system (2018). arXiv:1806.02798) proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020) propose a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In the present paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall (Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85, vol. 1204, pp. 447–464. Lecture Notes in Mathematics. Springer, Berlin (1986)). A ball configuration distributed as independent Bernoulli variables of parameter λ < 1∕2 is in correspondence with a simple random walk with negative drift 2λ − 1 and having infinitely many excursions over the local minima. In this case the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020). We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Gabrielli, Davide. Universita degli Studi dell'Aquila; ItaliaXIII Symposium on Probability and Stochastic ProcessesMéxicoUniversidad Nacional Autónoma de Méxic

Separation versus diffusion in a two species system

We consider a finite number of particles that move in ZZ as independent random walks. The particles are of two species that we call aa and bb. The rightmost aa-particle becomes a bb-particle at constant rate, while the leftmost bb-particle becomes aa-particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a nonlinear system of two PDE’s with free boundaries.Fil: De Masi, Anna. Università di L’Aquila; ItaliaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Quasi Stationary Distributions and Fleming - Viot Processes

This is an introduction to the volume dedicated to the meeting Interacting Random Systems coordinated by the author. Fil: Ferrari, Pablo Augusto. Universidad de Buenos Aires; Argentina. Universidade de Sao Paulo; Brasil. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Finite Cycle Gibbs Measures on Permutations of Zd

We consider Gibbs distributions on the set of permutations of Zd associated to the Hamiltonian H(σ ) := x V(σ (x) − x), where σ is a permutation and V : Zd → 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 μα concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct μα 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 continuoustime birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define τv as the shift permutation τv(x) = x + v. In the Gaussian case V =·2, we show that for each v ∈ Zd , μα v given by μα v ( f ) = μα[ f (τv·)] is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with τv boundary conditions. For a general potential V, we prove the existence of Gibbs measures μα v when α is bigger than some v-dependent value.Fil: Armendariz, Inés. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidade de Sao Paulo; BrasilFil: Groisman, Pablo Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Leonardi, Florencia Graciela. Universidade de Sao Paulo; Brasi

Phase Transition for Infinite Systems of Spiking Neurons

We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate 1 whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all presynaptic neurons since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to 0 and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate γ> 0. For this process we prove the existence of a value γc such that the system has one or two extremal invariant measures according to whether γ> γc or not.Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Galves, Antonio. Universidade de Sao Paulo; BrasilFil: Grigorescu, I.. University of Miami; Estados UnidosFil: Löcherbach, E.. Université Paris Seine; Franci

Gaussian random permutation and the boson point process

We construct an infinite volume spatial random permutation (χ,σ), where χ⊂ℝd is a point process and σ:χ→χ is a permutation (bijection), associated to the formal Hamiltonian H(χ,σ)=∑_x∈χ‖x−σ(x)‖2. The measures are parametrized by the density ρ of points and the temperature α. Feynman (1953) related spatial random permutations with boson systems and proposed that Bose-Einstein condensation occurs precisely when infinite cycles appear in the corresponding random permutation. Each finite cycle of σ induces a loop of points of χ. For ρ ≤ ρc we define (χ, σ) as a Poisson process of finite unrooted loops 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 double-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2007). For d ≥ 3 and ρ > ρc we define (χ, σ) as the superposition of independent realizations of the Gaussian loop soup at density ρc and the Gaussian random interlacements at density ρ − ρc and call it a Gaussian random permutation at density ρ and temperature α. The resulting measure is Gibbs for the Hamiltonian H and the point marginal χ has the same distribution as the boson point process introduced by Macchi (1975) in the subcritical case and by Tamura-Ito (2007) in the supercritical case. Bose-Einstein condensation occurs when the Gaussian random permutation exhibits infinite trajectories.Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Yuhjtman, Sergio Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

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∑_xV(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 (1) 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 ℤ^d with intensity ρ∈(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 ℝ^d with low enough intensity.Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Frevenza Maestrone, Nicolas Federico. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Cytokine Production but Lack of Proliferation in Peripheral Blood Mononuclear Cells from Chronic Chagas' Disease Cardiomyopathy Patients in Response to T. cruzi Ribosomal P Proteins

Background:Trypanosoma cruzi ribosomal P proteins, P2β and P0, induce high levels of antibodies in patients with chronic Chagas' disease Cardiomyopathy (CCC). It is well known that these antibodies alter the beating rate of cardiomyocytes and provoke apoptosis by their interaction with β1-adrenergic and M2-muscarinic cardiac receptors. Based on these findings, we decided to study the cellular immune response to these proteins in CCC patients compared to non-infected individuals.Methodology/Principal findings:We evaluated proliferation, presence of surface activation markers and cytokine production in peripheral blood mononuclear cells (PBMC) stimulated with P2β, the C-terminal portion of P0 (CP0) proteins and T. cruzi lysate from CCC patients predominantly infected with TcVI lineage. PBMC from CCC patients cultured with P2β or CP0 proteins, failed to proliferate and express CD25 and HLA-DR on T cell populations. However, multiplex cytokine assays showed that these antigens triggered higher secretion of IL-10, TNF-α and GM-CSF by PBMC as well as both CD4+ and CD8+ T cells subsets of CCC subjects. Upon T. cruzi lysate stimulation, PBMC from CCC patients not only proliferated but also became activated within the context of Th1 response. Interestingly, T. cruzi lysate was also able to induce the secretion of GM-CSF by CD4+ or CD8+ T cells.Conclusions/Significance:Our results showed that although the lack of PBMC proliferation in CCC patients in response to ribosomal P proteins, the detection of IL-10, TNF-α and GM-CSF suggests that specific T cells could have both immunoregulatory and pro-inflammatory potential, which might modulate the immune response in Chagas' disease. Furthermore, it was possible to demonstrate for the first time that GM-CSF was produced by PBMC of CCC patients in response not only to recombinant ribosomal P proteins but also to parasite lysate, suggesting the value of this cytokine to evaluate T cells responses in T. cruzi infection.Fil: Longhi, Silvia Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Investigaciones en Ingeniería Genética y Biología Molecular "Dr. Héctor N. Torres"; Argentina. Universidad de Buenos Aires. Facultad de Farmacia y Bioquímica; ArgentinaFil: Atienza, Augusto. Gobierno de la Ciudad de Buenos Aires. Hospital General de Agudos "Ramos Mejía"; ArgentinaFil: Perez Prados, Graciela. Gobierno de la Ciudad de Buenos Aires. Hospital General de Agudos "Juan A. Fernández"; ArgentinaFil: Buying, Alcinette. Torrey Pines Institute for Molecular Studies; Estados UnidosFil: Balouz, Virginia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Biotecnológicas. Universidad Nacional de San Martín. Instituto de Investigaciones Biotecnológicas; ArgentinaFil: Buscaglia, Carlos Andres. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Biotecnológicas. Universidad Nacional de San Martín. Instituto de Investigaciones Biotecnológicas; ArgentinaFil: Santos, Radleigh. Torrey Pines Institute for Molecular Studies; Estados UnidosFil: Tasso, Laura Mónica. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Investigaciones en Ingeniería Genética y Biología Molecular "Dr. Héctor N. Torres"; ArgentinaFil: Bonato, Ricardo. Gobierno de la Ciudad de Buenos Aires. Hospital General de Agudos "Ramos Mejía"; ArgentinaFil: Chiale, Pablo. Gobierno de la Ciudad de Buenos Aires. Hospital General de Agudos "Ramos Mejía"; ArgentinaFil: Pinilla, Clemencia. Torrey Pines Institute for Molecular Studies; Estados UnidosFil: Judkowski, Valeria A.. Torrey Pines Institute for Molecular Studies; Estados UnidosFil: Gomez, Karina Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Investigaciones en Ingeniería Genética y Biología Molecular "Dr. Héctor N. Torres"; Argentina. Universidad de Buenos Aires. Facultad de Farmacia y Bioquímica; Argentin