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

Rank Dependent Branching-Selection Particle Systems

We consider a large family of branching-selection particle systems. Thebranching rate of each particle depends on its rank and is given by a functionb defined on the unit interval. There is also a killing measure D supportedon the unit interval as well. At branching times, a particle is chosen amongall particles to the left of the branching one by sampling its rank accordingto D. The measure D is allowed to have total mass less than one, whichcorresponds to a positive probability of no killing. Between branching times,particles perform independent Brownian Motions in the real line. This settingincludes several well known models like Branching Brownian Motion (BBM),N-BBM, rank dependent BBM, and many others. We conjecture a scaling limit forthis class of processes and prove such a limit for a related class ofbranching-selection particle system. This family is rich enough to allow us touse the behavior of solutions of the limiting equation to prove the asymptoticvelocity of the rightmost particle under minimal conditions on b and D. Thebehavior turns out to be universal and depends only on b(1) and the totalmass of D. If the total mass is one, the number of particles in the systemN is conserved and the velocities vN converge to 2b(1)‾‾‾‾‾√. When thetotal mass of D is less than one, the number of particles in the system growsup in time exponentially fast and the asymptotic velocity of the rightmost oneis 2b(1)‾‾‾‾‾√ independently of the number of initial particles.Fil: 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: Soprano Loto, Nahuel. 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

Online matching for the multiclass stochastic block model

We consider the problem of sequential matching in a stochastic block model with several classes of nodes and generic compatibility constraints. When the probabilities of connections do not scale with the size of the graph, we show that under the NCOND condition, a simple max-weight type policy allows to attain an asymptotically perfect matching while no sequential algorithm attain perfect matching otherwise. The proof relies on a specific Markovian representation of the dynamics associated with Lyapunov techniques

Thermodynamics for spatially inhomogeneous magnetization and Young-Gibbs measures

We derive thermodynamic functionals for spatially inhomogeneous magnetization on a torus in the context of an Ising spin lattice model. We calculate the corresponding free energy and pressure (by applying an appropriate external field using a quadratic Kac potential) and show that they are related via a modified Legendre transform. The local properties of the infinite volume Gibbs measure, related to whether a macroscopic configuration is realized as a homogeneous state or as a mixture of pure states, are also studied by constructing the corresponding Young-Gibbs measures

Phase transition for dilute models with discrete spins and Young-Gibbs measures for the Ising model

Esta tesis contiene dos partes con un tema en común: en cada una de ellas, estudiamos diferentes modelos de mecánica estadística. En la primera parte, estudiamos modelos diluidos de vecinos próximos con espacio de espines finito, donde el grafo subyacente es un subgrafo aleatorio del reticulado d-dimensional. Más precisamente, proporcionamos condiciones suficientes y necesarias para que ocurra co-existencia de fases mediante técnicas de aglomerado aleatorio. En la segunda parte, estudiamos un modelo del tipo Ising con interacciones de vecinos próximos ferromagnéticas y potencial cuadrático del tipo Kac asociado a un campo externo no-homogéneo. En este caso, probamos que la energía libre y la presión existen y establecemos resultados de grandes desvíos y equivalencia de arreglos.This thesis contains two parts with one topic in common: in each one, we study different statistical-mechanical models. In the first part, we study dilute nearest-neighbour models with finite spin state, being the underlying graph a random subgraph of the d-dimensional lattice. More precisely, we give necessary and sufficient conditions for phase co-existence to occur via random-cluster techniques. In the second part, we study an Ising-type model with ferromagnetic nearest-neighbour interactions and quadratic Kac-type potential associated to an inhomogeneous external field. In this case, we prove that the free energy and the pressure exist and establish large deviation and equivalence of ensembles results.Fil:Soprano Loto, Nahuel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Turing Instability in a Model with Two Interacting Ising Lines: Non-equilibrium Fluctuations

This is the second of two articles on the study of a particle system model that exhibits a Turing instability type effect. About the hydrodynamic equations obtained in Capanna and Soprano (Markov Proc Relat Fields 23(3):401–420, 2017), we find conditions under which Turing instability occurs around the zero equilibrium solution. In this instability regime: for long times at which the process is of infinitesimal order, we prove that the non-equilibrium fluctuations around the hydrodynamic limit are Gaussian; for times converging to the critical time defined as the one at which the process starts to be of finite order, we prove that the ±1-Fourier modes are uniformly away from zero.Fil: Capanna, Monia. 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. Università degli Studi dell’Aquila; ItaliaFil: Soprano Loto, Nahuel. Gran Sasso Science Institute; Italia. 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

Turing instability in a model with two interacting Ising lines: hydrodynamic limit

This is the first of two articles on the study of a particle system model that exhibits  a Turing instability type effect. The model is based on two discrete lines (or toruses) with Ising spins, that evolve according to a continuous time Markov process defined in terms of macroscopic Kac potentials and local interactions. For fixed time, we prove that the density fields weakly converge to the solution of a system of partial differential equations involving convolutions. The presence of local interactions results in the lack of propagation of chaos, reason why the hydrodynamic limit cannot be obtained from previous results.Fil: Capanna, Monia. Università Degli Studi Dell'aquila; Italia. 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: Soprano Loto, Nahuel. 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. Gran Sasso Science Institute; Itali

Generalized max-weight policies in stochastic matching

19 pages, 4 figuresInternational audienceWe consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, i.e. leave the system before being matched. Using specific properties of the max-weight policy, we construct several Lyapunov functions, including a simple quadratic one. This allows us to establish stability results, to construct bounds for the stationary mean and variances of the total amount of customers in the system, and to prove exponential convergence speed towards the stationary measure. We finally illustrate some of these results using simulations on toy examples