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Ricercatore t.d. art. 24 c. 3 lett. B presso: Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Carinci, G; Giardinà, Cristian; Redig, F

We consider two particles performing continuous-time nearest neighbor random walk on Z and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace-Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions and sticky Brownian motions.In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime. We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterise the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.

2020 - Stationary States in Infinite Volume with Non-zero Current [Articolo su rivista]
Carinci, G.; Giardina, C.; Presutti, E.

We study the Ginzburg–Landau stochastic models in infinite domains with some special geometry and prove that without the help of external forces there are stationary measures with non-zero current in three or more dimensions.

2019 - Orthogonal duality of Markov processes and unitary symmetries [Articolo su rivista]
Carinci, Gioia; Franceschini, Chiara; Giardina', Cristian; Groenevelt, Wolter Godfried Mattijs; Redig, Frank

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.

2018 - Quantitative Boltzmann–Gibbs Principles via Orthogonal Polynomial Duality [Articolo su rivista]
Ayala, M.; Carinci, G.; Redig, F.

We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann–Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context (local equilibrium). For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the n particle dynamics.

2016 - A generalized asymmetric exclusion process with Uq(sl2) stochastic duality [Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Redig, Frank; Sasamoto, Tomohiro

We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by (Formula presented.) and where at most (Formula presented.) particles per site are allowed. The process is constructed from a (Formula presented.)-dimensional representation of a quantum Hamiltonian with (Formula presented.) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several self-duality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure.

2016 - Asymmetric Stochastic Transport Models with Uq(su(1,1)) Symmetry [Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Redig, Frank; Tomohiro, Sasamoto

By using the algebraic construction outlined in Carinci et al. (arXiv:1407.3367, 2014), we introduce several Markov processes related to the (Formula presented.) quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the and which turns out to have the Symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

2016 - Free boundary problems in PDEs and particle systems [Monografia/Trattato scientifico]
Gioia, Carinci; Anna De Masi, ; Cristian, Giardina'; Errico, Presutti

In this volume a theory for models of transport in the presence of a free boundary is developed.Macroscopic laws of transport are described by PDE's. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analysed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases.All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms.In general researchers interested in the relations between PDE’s and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.

2015 - Dualities in population genetics: A fresh look with new dualities [Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Giberti, Claudio; Frank, Redig

We apply our general method of duality, introduced in [15], to models of population dynamics. The classical dualities between forward and ancestral processes can be viewed as a change of representation in the classical creation and annihilation operators, both for diffusions dual to coalescents of Kingman’s type, as well as for models with finite population size. Next, using SU(1, 1) raising and lowering operators, we find new dualities between the Wright-Fisher diffusion with d types and the Moran model, both in presence and absence of mutations. These new dualities relates two forward evolutions. From our general scheme we also identify self-duality of the Moran model.

2014 - Hydrodynamic limit in a particle system with topological interactions [Articolo su rivista]
Carinci, Gioia; Anna De, Masi; Giardina', Cristian; Errico, Presutti

We study a system of particles in the interval [0, \eps^{ −1}] ∩ Z, \eps^{−1} a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate j (j > 0) and removed at same rate from the rightmost occupied site. The removal mechanism is therefore of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields \eps \sum φ(\eps x) ξ_{\eps^{-2}t} (x) (φ a test function, ξ_{t}(x) the number of particles at site x at time t) concentrates in the limit t → 0 on the deterministic value R \int φ ρ_t, ρ_t interpreted as the limit density at time t. We characterize the limit ρ_t as a weak solution in terms of barriers of a limit free boundary problem.

2014 - Super-Hydrodynamic Limit in Interacting Particle Systems [Articolo su rivista]
Carinci, Gioia; Anna De, Masi; Giardina', Cristian; Errico, Presutti

This paper is a follow-up of the work initiated in [3], where it has been investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale and show that they are given by a family of linear macroscopic profiles whose parameters are determined by the current reservoirs and the system mass. Then we prove the existence of a superhyrdrodynamic time scale, beyond the hydrodynamic one. On this larger time scale the system mass fluctuates and correspondingly the macroscopic profile of the system randomly moves within the family of linear profiles, with the randomness of a Brownian motion.

2013 - Duality for Stochastic Models of Transport [Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Giberti, Claudio; F., Redig

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particle densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneously redistributed among sites. The results shows that duality is a versatile tool for analyzing stochastic models of transport, while the analysis in the literature has been so far limited to particular instances. Longrange correlations naturally emerge as a result of the interaction of dual particles at the microscopic level and the explicit computations of covariances match, in the scaling limit, the predictions of the macroscopic fluctuation theory.

2013 - Langevin Dynamics with a Tilted Periodic Potential [Articolo su rivista]
Carinci, G.; Luckhaus, S.

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity γ and subject to a further external field α. For a suitable choice of the parameters α and γ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale. © 2013 Springer Science+Business Media New York.

2013 - Random hysteresis loops [Articolo su rivista]
Carinci, G.

Dynamical hysteresis is a phenomenon which arises in ferromagnetic systems below the critical temperature as a response to adiabatic variations of the external magnetic field. We study the problem in the context of the mean-field Ising model with Glauber dynamics, proving that for frequencies of the magnetic field oscillations of order N -2/3, N the size of the system, the "critical" hysteresis loop becomes random. © 2013 Association des Publications de l'Institut Henri Poincaré.

2012 - Nonconventional averages along arithmetic progressions and lattice spin systems [Articolo su rivista]
Carinci, G.; Chazottes, J. -R.; Giardina, C.; Redig, F.

We study the so-called nonconventional averages in the context of lattice spin systems, or equivalently random colorings of the integers. For i.i.d. colorings, we prove a large deviation principle for the number of monochromatic arithmetic progressions of size two in the box [1,N]∩N, as N→∞, with an explicit rate function related to the one-dimensional Ising model.For more general colorings, we prove some bounds for the number of monochromatic arithmetic progressions of arbitrary size, as well as for the maximal progression inside the box [1,N]∩N.Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.