
Cristian GIARDINA'
Professore Ordinario Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exScienze Comunicazione

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2024
 Density Fluctuations for the MultiSpecies Stirring Process
[Articolo su rivista]
Casini, F.; Giardinà, C.; Redig, F.
abstract
We study the density fluctuations at equilibrium of the multispecies stirring process, a natural multitype generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinitedimensional OrnsteinUhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multispecies stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.
2024
 Duality for the multispecies stirring process with open boundaries
[Articolo su rivista]
Casini, Francesco; Frassek, Rouven; Giardinà, Cristian
abstract
We study the stirring process with N  1 species on a generic graph G = (V, E) with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case N = 2. We prove the existence of a dual process defined on an extended graph (G) over bar = ((V) over tilde, (E) over bar) which includes additional sites in (V) over bar \V where dual particles get absorbed in the longtime limit. We thus obtain a characterization of the nonequilibrium steady state of the boundarydriven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the onedimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying gl(N) symmetry and the matrix product ansatz. Thus one gets a closedformula for (longranged) correlations and for the nonequilibrium stationary measure. Extensions beyond this integrable setup are also discussed.
2024
 Solvable Stationary Non Equilibrium States
[Articolo su rivista]
Carinci, G.; Franceschini, C.; Gabrielli, D.; Giardinà, C.; Tsagkarogiannis, D.
abstract
We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135171, 2020). By combining duality and integrability the authors of (Frassek and Giardina in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications.
2023
 Integrable heat conduction model
[Articolo su rivista]
Franceschini, Chiara; Frassek, Rouven; Giardina, Cristian
abstract
We consider a stochastic process of heat conduction where energy is redistributed along a chain between nearest neighbor sites via an improper beta distribution. Similar to the wellknown KipnisMarchioroPresutti (KMP) model, the finite chain is coupled at its ends with two reservoirs that break the conservation of energy when working at different temperatures. At variance with KMP, the model considered here is integrable and one can write in a closed form the $n$point correlation functions of the nonequilibrium steady state. As a consequence of the exact solution one can directly prove that the system is in a `local equilibrium' and described at the macroscale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with noncompact spins. The algebraic formulation of the model allows to interpret its duality relation with a purely absorbing particle system as a change of representation.
2023
 Uphill in ReactionDiffusion Multispecies Interacting Particles Systems
[Articolo su rivista]
Casini, F.; Giardinà, C.; Vernia, C.
abstract
We study reactiondiffusion processes with multispecies particles and hardcore interaction. We add boundary driving to the system by means of external reservoirs which inject and remove particles, thus creating stationary currents. We consider the condition that the time evolution of the average occupation evolves as the discretized version of a system of coupled diffusive equations with linear reactions. In particular, we identify a specific oneparameter family of such linear reactiondiffusion systems where the hydrodynamic limit behaviour can obtained by means of a dual process. We show that partial uphill diffusion is possible for the discrete particle systems on the lattice, whereas it is lost in the hydrodynamic limit.
2022
 Annealed inhomogeneities in random ferromagnets
[Articolo su rivista]
Hao Can, Van; Giardina', Cristian; Giberti, Claudio; van der Hofstad, Remco
abstract
We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (ErdősRényi) or prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite because then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the meanfield exponents if the degree distribution has finite fourth moment and powerlawdependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random independent and identically distributed degrees washes away the universality class with powerlaw critical exponents.
2022
 Annealed Ising model on configuration models
[Articolo su rivista]
Can, Van Hao; Giardinà, Cristian; Giberti, Claudio; van der Hofstad, Remco
abstract
In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we
take the average on both sides of the ratio defining the Boltzmann–Gibbs measure of the Ising model. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.
2022
 Exact solution of an integrable nonequilibrium particle system
[Articolo su rivista]
Frassek, Rouven; Giardinà, Cristian
abstract
We consider the integrable family of symmetric boundarydriven interacting particle systems that arise from the noncompact XXX Heisenberg model in one dimension with open boundaries. In contrast to the wellknown symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the nonequilibrium steadystate can be written in the closed form for each N. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: (i) the introduction of a dual absorbing process reducing the problem to a finite number of particles and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Longrange correlations are computed in the finitevolume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A byproduct of the solution is the algebraic construction of a direct mapping between the nonequilibrium steady state and the equilibrium reversible measure.
2022
 Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer
[Articolo su rivista]
Floreani, Simone; Giardinà, Cristian; Frank den Hollander, ; Nandan, Shubamoy; Redig, Frank
abstract
This paper considers three classes of interacting particle systems on Z: independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the type of particle) between 1 (fast particles) and ϵ∈ [0 , 1] (slow particles). The switch between the two jump rates happens at rate γ∈ (0 , ∞). In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by N 1, time by N2, the switching rate by N 2, and letting N→ ∞. The limit equations for the macroscopic densities associated to the fast and slow particles is the wellstudied double diffusivity model. This system of reactiondiffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick’s law. In order to investigate the microscopic outofequilibrium properties, we analyse the system on [N] = { 1 , … , N} , adding boundary reservoirs at sites 1 and N of fast and slow particles, respectively. Inside [N] particles move as before, but now particles are injected and absorbed at sites 1 and N with prescribed rates that depend on the particle type. We compute the steadystate density profile and the steadystate current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a singletype particle system, is a violation of Fick’s law made possible by the switching between types. We rescale the microscopic steadystate density profile and steadystate current and obtain the steadystate solution of a boundaryvalue problem for the double diffusivity model.
2021
 Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph
[Articolo su rivista]
Giardina', Cristian; Giberti, Claudio; Magnanini, Elena
abstract
We study the pressure of the “edgetriangle model”, which is equivalent to the cumulant
generating function of triangles in the Erdös–Rényi random graph. The investigation involves
a population dynamics method on finite graphs of increasing volume, as well as a discretization
of the graphon variational problem arising in the infinite volume limit. As a result, we
locate a curve in the parameter space where a onestep replica symmetry breaking transition
occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon
with a structure very close to the one of an equibipartite graph.
2021
 Consistent particle systems and duality
[Articolo su rivista]
Carinci, G.; Giardina, C.; Redig, F.
abstract
We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the KipnisMarchioroPresutti model. Consistent systems are such that the distribution obtained by first evolving n particles and then removing a particle at random is the same as the one given by a random removal of a particle at the initial time followed by evolution of the remaining n − 1 particles. In this paper we discuss two main results. Firstly, we show that, for reversible systems, the property of consistency is equivalent to selfduality, thus obtaining a novel probabilistic interpretation of the selfduality property. Secondly, we show that consistent particle systems satisfy a set of recursive equations. This recursions implies that factorial moments of a system with n particles are linked to those of a system with n − 1 particles, thus providing substantial information to study the dynamics. In particular, for a consistent system with absorption, the particle absorption probabilities satisfy universal recurrence relations. Since particle systems with absorption are often dual to boundarydriven nonequilibrium systems, the consistency property implies recurrence relations for expectations of correlations in nonequilibrium steady states. We illustrate these relations with several examples.
2021
 Duality in quantum transport models
[Articolo su rivista]
Frassek, R.; Giardina', C.; Kurchan, J.
abstract
We develop the ‘duality approach’, that has been extensively studied for classical models of transport, for quantum systems in contact with a thermal ‘Lindbladian’ bath. The method provides (a) a mapping of the original model to a simpler one, containing only a few particles and (b) shows that any dynamic process of this kind with generic baths may be mapped onto one with equilibrium baths. We exemplify this through the study of a particular model: the quantum symmetric exclusion process introduced in [1]. As in the classical case, the whole construction becomes intelligible by considering the dynamical symmetries of the problem.
2020
 Duality and hidden equilibrium in transport models
[Articolo su rivista]
Frassek, R.; Giardina', C.; Kurchan, J.
abstract
A large family of diffusive models of transport that have been considered in the past years admit a transformation into the same model in contact with an equilibrium bath. This mapping holds at the full dynamical level, and is independent of dimension or topology. It provides a good opportunity to discuss questions of time reversal in out of equilibrium contexts. In particular, thanks to the mapping one may define the free energy in the nonequilibrium states very naturally as the (usual) free energy of the mapped system.
2020
 EXACT FORMULAS FOR TWO INTERACTING PARTICLES AND APPLICATIONS IN PARTICLE SYSTEMS WITH DUALITY
[Articolo su rivista]
Carinci, G; Giardina', Cristian; Redig, F
abstract
We consider two particles performing continuoustime 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 LaplaceFourier transform of the transition probabilities of the twoparticle 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 selfdual, in the condensation regime. We thus obtain two results. First, by computing the timedependent covariance of the particle occupation number at two lattice sites we characterise the timedependent 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
 Noncompact quantum spin chains as integrable stochastic particle processes
[Articolo su rivista]
Frassek, R.; Giardinà, C; Kurchan, J
abstract
In this paper we discuss a family of models of particle and energy diffusion on a onedimensional lattice, related to those studied previously in Sasamoto and Wadati (Phys Rev E 58:4181–4190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2017) and Povolotsky (J Phys A 46(46):465205, 2013) in the context of KPZ universality class. We show that they may be mapped onto an integrable (2) Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in nonequilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a “dual model” of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of =4 super Yang–Mills theory (SYM), in imaginarytime. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the (21) superstring that has been derived directly from =4 SYM.
2020
 Stationary States in Infinite Volume with Nonzero Current
[Articolo su rivista]
Carinci, G.; Giardina, C.; Presutti, E.
abstract
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 nonzero current in three or more dimensions.
2020
 The NonEquilibrium Ising Model in Two Dimensions: a Numerical Study
[Articolo su rivista]
Giardina, Cristian
abstract
In this paper, we study the boundarydriven ferromagnetic Ising model in two dimensions. In this nonequilibrium setting, in the low temperature region, the Ising model has phase separation in the presence of a current. We investigate, by means of numerical simulations, Kawasaki dynamics with magnetization reservoirs. The results show that, in the stationary nonequilibrium state, the Ising model may have uphill diffusion and magnetization profiles with three discontinuities. These results complement the results of a previous paper by Colangeli, Giberti, Vernia and the present author [9]. They also allow to state a full picture of the hydrodynamic limit.
2019
 Orthogonal duality of Markov processes and unitary symmetries
[Articolo su rivista]
Carinci, Gioia; Franceschini, Chiara; Giardina', Cristian; Groenevelt, WOLTER GODFRIED MATTIJS; Redig, Frank
abstract
We study selfduality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal selfdualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the BakerCampbellHausdorff 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 nonorthogonal duality functions.
2019
 Stochastic Duality and Orthogonal Polynomials
[Capitolo/Saggio]
Franceschini, C.; Giardina', C.
abstract
For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal polynomial) can be studied via expectations with respect to the dual process (which evolves the index of the polynomial). The set of processes include interacting particle systems, such as the exclusion process, the inclusion process and independent random walkers, as well as interacting diffusions and redistribution models of Kipnis–Marchioro–Presutti type. Duality functions are given in terms of classical orthogonal polynomials, both of discrete and continuous variable, and the measure in the orthogonality relation coincides with the process stationary measure.
2018
 Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees
[Articolo su rivista]
Dommers, Sander; Giardinà, Cristian; Giberti, Claudio; Hofstad, Remco Van Der
abstract
We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs.
2018
 Nonequilibrium twodimensional Ising model with stationary uphill diffusion
[Articolo su rivista]
Colangeli, Matteo; Giardinà, Cristian; Giberti, Claudio; Vernia, Cecilia
abstract
Usually, in a nonequilibrium setting, a current brings mass from the highest density regions to the lowest density ones. Although rare, the opposite phenomenon (known as “uphill diffusion”) has also been observed in multicomponent systems, where it appears as an artificial effect of the interaction among components. We show here that uphill diffusion can be a substantial effect, i.e., it may occur even in single component systems as a consequence of some external work. To this aim we consider the twodimensional ferromagnetic Ising model in contact with two reservoirs that fix, at the left and the right boundaries, magnetizations of the same magnitude but of opposite signs.We provide numerical evidence that a class of nonequilibrium steady states exists in which, by tuning the reservoir magnetizations, the current in the system changes from “downhill” to “uphill”. Moreover, we also show that, in such nonequilibrium setup, the current vanishes when the reservoir magnetization attains a value approaching, in the large volume limit, the magnetization of the equilibrium dynamics, thus establishing a relation between equilibrium and nonequilibrium properties.
2018
 SelfDuality of Markov Processes and Intertwining Functions
[Articolo su rivista]
Franceschini, Chiara; Giardina', Cristian; Wolter, Groenevelt
abstract
We present a theorem which elucidates the connection between selfduality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the selfduality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The selfdual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitarily equivalent. As a consequence, in the discrete setting selfduality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.
2017
 Metastability in the reversible inclusion process
[Articolo su rivista]
Bianchi, Alessandra; Dommers, Sander; Giardinà, Cristian
abstract
We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph SS with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices S⋆⊆SS⋆⊆S. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to S⋆S⋆ is irreducible, then there exists a single timescale for the condensate motion. In this case we compute this typical timescale and characterize the law of the (properly rescaled) limiting process. If the restriction of the random walk to S⋆S⋆ has several connected components, a metastability scenario with multiple timescales emerges. We prove such a scenario, involving two additional timescales, in a onedimensional setting with two metastable states and nearestneighbor jumps.
2016
 A generalized asymmetric exclusion process with Uq(sl2) stochastic duality
[Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Redig, Frank; Sasamoto, Tomohiro
abstract
We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a onedimensional 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 groundstate transformation. After showing basic properties of the process ASEP(q, j), we prove selfduality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the selfduality property we compute the first qexponential 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
 Advances in disordered systems, random processes and some applications
[Monografia/Trattato scientifico]
Contucci, P.; Giardinà, Cristian
abstract
This book offers a unified perspective on the study of complex systems for scholars of various disciplines, including mathematics, physics, computer science, biology, economics and social science. The contributions, written by leading scientists, cover a broad set of topics, including new approaches to data science, the connection between scaling limits and conformal field theories, and new ideas on the Legendre duality approach in statistical mechanics of disordered systems. The volume moreover explores results on extreme values of correlated random variables and their connection with the Riemann zeta functions, the relation between diffusion phenomena and complex systems, and the Brownian web, which appears as the universal scaling limit of several probabilistic models. Written for researchers from a broad range of scientific fields, this text examines a selection of recent developments in complex systems from a rigorous perspective.
2016
 Annealed central limit theorems for the Ising model on random graphs
[Articolo su rivista]
Giardina', Cristian; Giberti, Claudio; van der Hofstad, Remco; Prioriello, Maria Luisa
abstract
The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled
by $sqrt{N}$ of Ising models on random graphs. More precisely, we consider the general rank1 inhomogeneous random graph (or generalized random graph), the 2regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature $0le eta^{mathrm{an}}_c < infty$ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature $eta$ and external magnetic field $B$ for which either $eta<eta^{mathrm{an}}_c$ and $B=0$, or $eta>0$ and $B
eq 0$. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters $eta$ and $B$, because phase transitions do not exist for these systems as they are closely related to onedimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous CurieWeiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical onedimensional Ising model.
2016
 Asymmetric Stochastic Transport Models with Uq(su(1,1)) Symmetry
[Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Redig, Frank; Tomohiro, Sasamoto
abstract
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 selfdual; (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]
Carinci, Gioia; De Masi, Anna; Giardina', Cristian; Presutti, Errico
abstract
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.
2016
 Ising Critical Behavior of Inhomogeneous CurieWeiss Models and Annealed Random Graphs
[Articolo su rivista]
Dommers, Sander; Giardina', Cristian; Giberti, Claudio; van der Hofstad, Remco; Prioriello, Maria Luisa
abstract
We study the critical behavior for inhomogeneous versions of the CurieWeiss model, where the coupling constant Jij(β) for the edge ij on the complete graph is given by Jij(β) = βwiwj/ (∑ k∈[N]wk). We call the product form of these couplings the rank1 inhomogeneous CurieWeiss model. This model also arises [with inverse temperature β replaced by sinh (β) ] from the annealed Ising model on the generalized random graph. We assume that the vertex weights (wi)i∈[N] are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a nonclassical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) CurieWeiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent τ with τ∈ (3 , 5) , then the critical exponents depend sensitively on τ. In addition, at criticality, the total spin SN satisfies that SN/ N(τ2)/(τ1) converges in law to some limiting random variable whose distribution we explicitly characterize.
2016
 Preface
[Capitolo/Saggio]
Contucci, P.; Giardina', C.
abstract
2015
 Dualities in population genetics: A fresh look with new dualities
[Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Giberti, Claudio; Frank, Redig
abstract
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 WrightFisher 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 selfduality of the Moran model.
2015
 Quenched Central Limit Theorems for the Ising Model on Random Graphs
[Articolo su rivista]
Giardina', Cristian; Giberti, Claudio; van der Hofstad, Remco; Prioriello, MARIA LUISA
abstract
Themain goal of the paper is to prove central limit theorems for the magnetization
rescaled by the square root of N for the Ising model on random graphs with N vertices.Both random quenched
and averaged quenched measures are considered.We work in the uniqueness regime β > βc
or β > 0 and B not equal to 0, where β is the inverse temperature, βc is the critical inverse temperature
and B is the external magnetic field. In the random quenched setting our results apply to
general treelike random graphs (as introduced by Dembo, Montanari and further studied by
Dommers and the first and third author) and our proof follows that of Ellis in Z^d. For the
averaged quenched setting, we specialize to two particular random graph models, namely
the 2regular configuration model and the configuration model with degrees 1 and 2. In
these cases our proofs are based on explicit computations relying on the solution of the one
dimensional Ising models
2015
 Spatial fluctuation theorem
[Articolo su rivista]
PEREZ ESPIGARES, Carlos; Redig, Frank; Giardina', Cristian
abstract
For nonequilibrium systems of interacting particles and for interacting diffusions in ddimensions, a novel fluctuation relation is derived. The theorem
establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of
spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti–Cohen fluctuation theorem related to the timereversal symmetry.
This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
2014
 Hydrodynamic limit in a particle system with topological interactions
[Articolo su rivista]
Carinci, Gioia; Anna De, Masi; Giardina', Cristian; Errico, Presutti
abstract
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
 Ising Critical Exponents on Random Trees and Graphs
[Articolo su rivista]
Sander, Dommers; Giardina', Cristian; Remco van der, Hofstad
abstract
We study the critical behavior of the ferromagnetic Ising model on random trees as well
as socalled locally treelike random graphs. We pay special attention to trees and graphs
with a powerlaw offspring or degree distribution whose tail behavior is characterized by its
powerlaw exponent > 2. We show that the critical temperature of the Ising model equals
the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree
distribution. In particular, the inverse critical temperature equals zero when ∈ (2, 3] where
this mean equals infinity.
We further study the critical exponents , and
, describing how the (root) magnetiza
tion behaves close to criticality. We rigorously identify these critical exponents and show that
they take the values as predicted by Dorogovstev, et al. [9] and Leone et al. [17]. These values
depend on the powerlaw exponent , taking the meanfield values for > 5, but different
values for ∈ (3, 5).
2014
 SuperHydrodynamic Limit in Interacting Particle Systems
[Articolo su rivista]
Carinci, Gioia; Anna De, Masi; Giardina', Cristian; Errico, Presutti
abstract
This paper is a followup 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
 Antiferromagnetic Potts Model on the ErdősRényi Random Graph
[Articolo su rivista]
Pierluigi, Contucci; Sander, Dommers; Giardina', Cristian; Shannon, Starr
abstract
We study the antiferromagnetic Potts model on the Poissonian ErdősRényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature secondmoment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of DerridaRuelle probability cascades) and from an entropy positivity argument.
2013
 Duality for Stochastic Models of Transport
[Articolo su rivista]
Carinci, Gioia; Giardina', Cristian; Giberti, Claudio; F., Redig
abstract
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 socalled 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
 Interaction Flip Identities for non Centered Spin Glasses
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio
abstract
We consider spin glass models with noncentered interactions and investigate the
effect, on the random free energies, of flipping the interaction in a subregion
of the entire volume. A fluctuation bound obtained by martingale methods produces,
with the help of integration by parts technique, a family of polynomial
identities involving overlaps and magnetizations.
2012
 Nonconventional averages along arithmetic progressions and lattice spin systems
[Articolo su rivista]
Carinci, G.; Chazottes, J. R.; Giardina, C.; Redig, F.
abstract
We study the socalled 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 onedimensional 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.
2012
 Perspectives on spin glasses
[Monografia/Trattato scientifico]
Pierluigi, Contucci; Giardina', Cristian
abstract
Presenting and developing the theory of spin glasses as a prototype for complex systems, this book is a rigorous and uptodate introduction to their properties. The book combines a mathematical description with a physical insight of spin glass models. Topics covered include the physical origins of those models and their treatment with replica theory; mathematical properties like correlation inequalities and their use in the thermodynamic limit theory; main exact solutions of the mean field models and their probabilistic structures; and the theory of the structural properties of the spin glass phase such as stochastic stability and the overlap identities. Finally, a detailed account is given of the recent numerical simulation results and properties, including overlap equivalence, ultrametricity and decay of correlations. The book is ideal for mathematical physicists and probabilists working in disordered systems.
2012
 Structural spinglass identities from a stability property: an explicit derivation
[Relazione in Atti di Convegno]
Contucci, Pierluigi; Giardina', Cristian; Giberti, Claudio
abstract
In this paper a recent extension (P.Contucci, C.Giardina', C.Giberti, EPL.96, 17003 (2011)) of the stochastic stability property ( M.Aizenman, P.Contucci, Journal of Statistical Physics, Vol.92, N. 5/6, 765783, (1998)) is analyzed
and shown to lead to the Ghirlanda Guerra identities for Gaussian spin glass models.
The result is explicitly obtained by integration by parts techinque.
2011
 Interface Energy in the EdwardsAnderson Model
[Articolo su rivista]
Pierluigi, Contucci; Giardina', Cristian; Giberti, Claudio; Giorgio, Parisi; Vernia, Cecilia
abstract
We numerically investigate the spin glass energy interface problem in three dimensions. We analyze the energy cost of changing the overlap from −1 to +1 at one boundary of two coupled systems (in the other boundary the overlap is kept fixed to +1). We implement a parallel tempering algorithm that simulates finite temperature systems and works with both cubic lattices and parallelepiped with fixed aspect ratio. We find results consistent with a lower critical dimension D c =2.5. The results show a good agreement with the mean field theory predictions.
2011
 Simulating Rare Events in Dynamical Processes
[Articolo su rivista]
Giardina', Cristian; Jorge, Kurchan; Vivien, Lecomte; Julien, Tailleur
abstract
Atypical, rare trajectories of dynamical systems are important: they are often the paths for chemical reactions, the haven of (relative) stability of planetary systems, the rogue waves that are detected in oil platforms, the structures that are responsible for intermittency in a turbulent liquid, the active regions that allow a supercooled liquid to flow…. Simulating them in an efficient, accelerated way, is in fact quite simple.In this paper we review a computational technique to study such rare events in both stochastic and Hamiltonian systems. The method is based on the evolution of a family of copies of the system which are replicated or killed in such a way as to favor the realization of the atypical trajectories. We illustrate this with various examples.
2011
 Stability of the Spin Glass Phase under Perturbations
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio
abstract
We introduce and prove a novel linear response stability theory for spin glasses. The new stability under suitable perturbation of the equilibrium state implies the whole set of structural identities that characterize the spin glass phase.
2010
 Correlation Inequalities for Interacting Particle Systems with Duality
[Articolo su rivista]
Giardina', Cristian; Frank, Redig; Kyamars, Vafayi
abstract
We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other—a process which we call here the symmetric inclusion process (SIP)—or repel each other—a generalized version of the wellknown symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction,—the socalled Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.
2010
 Ising models on powerlaw random graphs
[Articolo su rivista]
Sander, Dommers; Giardina', Cristian; Remco van der, Hofstad
abstract
We study a ferromagnetic Ising model on random graphs with a powerlaw degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent τ>2), for which the random graph has a treelike structure. For this, we closely follow the analysis by Dembo and Montanari (Ann. Appl. Probab. 20(2):565–592, 2010) which assumes finite variance degrees (τ>3), adapting it when necessary and also simplifying it when possible. Our results also apply in cases where the degree distribution does not obey a power law.We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy.
2010
 Modelling Complex Systems with Statistical Mechanics: The Computational Approach
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio; Vernia, Cecilia
abstract
Realworld phenomena are often described by complex systems with competitive and cooperative behaviour. Such systems, as much as the described phenomena, are hard to understand in a scientific perspective mainly due to the lack of general exact solutions. For cases like this, the computational sciences provide a very useful virtual laboratory. The case of disordered systems is an example of scientific computing techniques being used to test theoretical predictions and uncover new phenomena that remain unreachable by traditional analytical methods.
2009
 Duality and hidden symmetries in interacting particle systems
[Articolo su rivista]
Giardina', Cristian; F., Redig; K., Vafayi
abstract
In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the "hidden" symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the KipnisMarchioroPresutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.
2009
 InteractionFlip Identities in Spin Glasses
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio
abstract
We study the properties of fluctuation for the free energies and internal energies of two spinglass systems that differ for having some set of interactions flipped. We show that their difference has avariance that grows like the volumeof the flipped region. Using a new interpolation method,which extends to the entire circle the standard interpolation technique, we show by integration by parts that the bound imply new overlap identities for the equilibrium state. As a side result the case of the noninteracting random field is analyzed and the triviality of its overlap distribution proved.
2009
 Matching with shift for onedimensional Gibbs measures
[Articolo su rivista]
Giardina', Cristian; P., Collet; F., Redig
abstract
We consider matching with shifts for Gibbsian sequences. We prove that the maximal overlap behaves as c log n, where c is explicitly identified in terms of the thermodynamic quantities (pressure) of the underlying potential. Our approach is based on the analysis of the first and second moment of the number of overlaps of a given size. We treat both the case of equal sequences (and nonzero shifts) and independent sequences. © Institute of Mathematical Statistics, 2009.
2009
 Spin Glass Identities and the Nishimori Line
[Capitolo/Saggio]
P., Contucci; Giardina', Cristian; H., Nishimori
abstract
For a general spin glass model with asymmetric couplings we prove a family of identities involving expectations of generalized overlaps and magnetizations in the quenched state. Those identities hold pointwise in the Nishimori line and are reached at the rate of the inverse volume while, in the general case, they can be proved in integral average
2009
 Structure of correlations in three dimensional spin glasses
[Articolo su rivista]
P., Contucci; GIARDINA', Cristian; GIBERTI, Claudio; G., Parisi; VERNIA, Cecilia
abstract
We investigate the low temperature phase of the three dimensional EdwardAnderson model with Bernoulli random couplings. We show that, at a fixed value Q of the overlap, the model fulfills the clustering property: The connected correlation functions between two local overlaps have power law decay. Our findings are in agreement with the replica symmetry breaking theory and show that the overlap is a good order parameter. © 2009 The American Physical Society.
2009
 Thinking transport as a twist.
[Articolo su rivista]
Giardina', Cristian; J., Kurchan
abstract
The determination of the conductivity of a deterministic or stochastic classical system coupled to reservoirs at its ends can in general be mapped onto the problem of computing the stiffness (the 'energy' cost of twisting the boundaries) of a quantumlike system. The nature of the coupling to the reservoirs determines the details of the mechanical coupling of the torque at the ends. © 2009 Springer Science+Business Media, LLC.
2008
 Answer to Comment on "Ultrametricity in the EdwardsAnderson Model"
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio; G., Parisi; Vernia, Cecilia
abstract
In this paper we reply to a critical comment by T. Jorg and F. Krzakala to the Letter "Ultrametricity in the EdwardsAnderson Model" PRL 99, 057206 (2007). We show that the procedure developed in the aforementioned paper to detect ultrametricity is able to discriminate the nonultrametric behavior of the twodimensional EdwardsAnderson model from the ultrametric threedimensional one. Moreover, the interesting finding of Jorg and Krzakala that in the twodimensional EdwardsAnderson model three random configurations have ordered overlaps fulfilling the ultrametric distribution is discussed and an explanation of this phenomenon is proposed.
2008
 Introduction to Special Issue: Statistical Mechanics on Random Structures
[Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract
Introduction to Special Issue: Statistical Mechanics on Random Structures
2008
 Mathematics and Social Science: A Statistical Mechanics Approach to Immigration
[Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract
Is modern science able to study social matters like those related to immigration phenomena on solid mathematical grounds? Can we for instance determine cultural robustness and the causes behind abrupt changes from cultural legacies? Can we predict, cause or avoid swings? A novel approach is under investigation using the statistical mechanics formalism devised for the study of phase transitions in physics.
2008
 SPECIAL ISSUE 2008: STATISTICAL MECHANICS ON RANDOM STRUCTURES
[Direzione o Responsabilità Riviste]
P., Contucci; Giardina', Cristian
abstract
2007
 Duality and exact correlations for a model of heat conduction
[Articolo su rivista]
Giardina', Cristian; J., Kurchan; F., Redig
abstract
We study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits longrange correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process. © 2007 American Institute of Physics.
2007
 The GhirlandaGuerra identities.
[Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract
If the variance of a Gaussian spinglass Hamiltonian grows like the volume the model fulfills the GhirlandaGuerra identities in terms of the normalized Hamiltonian covariance. © Springer Science+Business Media, LLC 2007.
2007
 Ultrametricity in the EdwardsAnderson model.
[Articolo su rivista]
P., Contucci; GIARDINA', Cristian; GIBERTI, Claudio; G., Parisi; VERNIA, Cecilia
abstract
We test the property of ultrametricity for the spinglass threedimensional EdwardsAnderson model in zero magnetic field with numerical simulations up to 203 spins. We find an excellent agreement with the prediction of the mean field theory. Since ultrametricity is not compatible with a trivial structure of the overlap distribution, our result contradicts the droplet theory. © 2007 The American Physical Society.
2007
 Variational bounds for the generalized random energy model.
[Articolo su rivista]
Giardina', Cristian; S., Starr
abstract
We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's "broken replica symmetry bounds," and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new. © 2007 Springer Science+Business Media, LLC.
2006
 Comment on ``Both site and link overlap distributions are non trivial in 3dimensional Ising spin glasses'', condmat/0608535v2
[Working paper]
P., Contucci; Giardina', Cristian
abstract
We comment on recent numerical experiments by G.Hed and E.Domany [condmat/0608535v2] on the quenched equilibrium state of the EdwardsAnderson spin glass model. The rigorous proof of overlap identities related to replica equivalence shows that the observed violations of those identities on finite size systems must vanish in the thermodynamic limit. See also the successive version condmat/0608535v4
2006
 Direct evaluation of largedeviation functions.
[Articolo su rivista]
Giardina', Cristian; J., Kurchan; L., Peliti
abstract
We introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities, such as current or density, that are local in time. The largedeviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values. We illustrate the method with the current fluctuations of the Totally Asymmetric Exclusion Process and with the work distribution of a driven Lorentz gas. © 2006 The American Physical Society.
2006
 Overlap equivalence in the EdwardsAnderson model
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio; Vernia, Cecilia
abstract
We study the relative fluctuations of the link overlap and the square standard overlap in the threedimensional Gaussian EdwardsAnderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e., deltalike distributed), then the other does also, and as a consequence, the "trivialnontrivial" picture should be dismissed. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the EdwardsAnderson model. © 2006 The American Physical Society.
2006
 Relative entropy and waiting times for continuoustime Markov processes
[Articolo su rivista]
J. R., Chazottes; Giardina', Cristian; F., Redig
abstract
For discretetime stochastic processes, there is a close connection between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot be straightforwardly extended to the continuoustime setting. Contrarily to the discretetime case one needs a reference measure on path space and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuoustime Markov processes with finite state space. A reference measure of special interest is the one associated to the timereversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almostsure convergence to relative entropy of the logarithm of waitingtimes ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle).
2005
 Factorization properties in the threedimensional EdwardsAnderson model
[Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract
We study the threedimensional Gaussian EdwardsAnderson model and find numerical evidence of a simple factorization law of the linkoverlaps distributions at large volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists, increasing the size of the system. Our results open new perspectives in the study of the two different overlaps emphasizing the importance of the concept of factorizationtriviality to distiniguish their role. © 2005 The American Physical Society.
2005
 Finding minima in complex landscapes: Annealed, greedy and reluctant algorithms.
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio; Vernia, Cecilia
abstract
We consider optimization problems for complex systems in which the cost function has a multivalleyed landscape. We introduce a new class of dynamical algorithms which, using a suitable annealing procedure coupled with a balanced greedyreluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the SherringtonKirkpatrick model of spinglasses. © World Scientific Publishing Company.
2005
 Interpolating greedy and reluctant algorithms
[Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio; Unguendoli, Francesco; Vernia, Cecilia
abstract
In a standard NPcomplete optimization problem, we introduce an interpolating algorithm between the quick decrease along the steepest descent direction (greedy dynamics) and a slow decrease close to the level curves (reluctant dynamics). We find that, for a fixed elapsed computer time, the best performance of the optimization is reached at a special value of the interpolation parameter, considerably improving the results of the pure cases of greedy and reluctant. © 2005 Taylor & Francis Group Ltd.
2005
 Spinglass stochastic stability: A rigorous proof
[Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract
We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spinglass quenched state. We show that stochastic stability holds in βaverage for both the SherringtonKirkpatrick model in terms of the square of the overlap function and for the EdwardsAnderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V <sup>1</sup>. As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applied to the thermal fluctuations only. Communicated by Jennifer Chayes. © 2005 Birkhäuser Verlag, Basel, Switzerland.
2005
 The Fourier law in a momentumconserving chain.
[Articolo su rivista]
Giardina', Cristian; J., Kurchan
abstract
We introduce a family of models for heat conduction with and without momentum conservation. They are analytically solvable in the high temperature limit and can also be efficiently simulated. In all cases the Fourier law is verified in one dimension. © IOP Publishing Ltd.
2004
 Numerical study of random superconductors
[Articolo su rivista]
Giardina', Cristian; J. M., Kosterlitz; N. V., Priezjev; N., Akino
abstract
The XY model with quenched random disorder is studied numerically at T = 0 by a defect scaling method as a model of a disordered superconductor. In 3D we find that, in the absence of screening, a vortex glass phase exists at low T for large disorder in 3D with stiffness exponent θ ≈ +0.31 and with finite screening and in 2D this phase does not exist. For weak disorder, a superconducting phase exists which we identify as a Bragg glass. In the presence of screened vortexvortex interactions, the vortex glass does not exist but the Bragg glass does. © 2004 Elsevier B.V. All rights reserved.
2004
 The thermodynamic limit for finite dimensional classical and quantum disordered systems
[Articolo su rivista]
P., Contucci; Giardina', Cristian; J., Pulé
abstract
We provide a very simple proof for the existence of the thermodynamic limit for the quenched specific pressure for classical and quantum disordered systems on a ddimensional lattice, including spin glasses. We develop a, method which relies simply on Jensen's inequality and which works for any disorder distribution with the only condition (stability) that the quenched specific pressure is bounded.
2003
 Energy landscape statistics of the random orthogonal model
[Articolo su rivista]
Giardina', Cristian; M., Degli Esposti; S., Graffi
abstract
The random orthogonal model (ROM) of MarinariParisiRitort [13, 14] is a model of statistical mechanics where the couplings among the spins are defined by a matrix chosen randomly within the orthogonal ensemble. It reproduces the most relevant properties of the Parisi solution of the SherringtonKirkpatrick model. Here we compute the energy distribution, and work out an estimate for the twopoint correlation function. Moreover, we show an exponential increase with the system size of the number of metastable states also for nonzero magnetic field.
2003
 Energydecreasing dynamics in meanfield spin models
[Articolo su rivista]
L., Bussolari; P., Contucci; M., Degli Esposti; Giardina', Cristian
abstract
We perform a statistical analysis of deterministic energydecreasing algorithms on meanfield spin models with a complex energy landscape, such as the Sine model and the SherringtonKirkpatrick model. We specifically address the following question: in the search for lowenergy configurations, which is more favorable (and in which sense)  a quick decrease along the gradient (greedy dynamics) or a slow decrease close to the level curves (reluctant dynamics)? Average time and wideness of the attraction basins are introduced for each algorithm, together with an interpolation among the two, and experimental results are presented for different system sizes. We found that while the reluctant algorithm performs better for a fixed number of trials, the two algorithms become basically equivalent for a given elapsed time due to the fact that the greedy algorithm has a shorter relaxation time which scales linearly with the system size compared to a quadratic dependence for the reluctant algorithm.
2003
 Multiple optimal solutions in the portfolio selection model with shortselling
[Articolo su rivista]
L., Bongini; M., Degli Esposti; Giardina', Cristian; A., Schianchi
abstract
In this paper an extension of the Lintner model [1] is considered: the problem of portfolio optimization is studied when shortselling is allowed through the mechanism of margin requirements. This induces a nonlinear constraint on the wealth. When interest on deposited margin is present, Lintner ingeniously solved the problem by recovering the unique optimal solution of the linear model (no margin requirements). In this paper an alternative and more realistic approach is explored: the nonlinear constraint is maintained but no interest is perceived on the money deposited against shortselling. This leads to a fully nonlinear problem which admits multiple and unstable solutions very different among themselves but corresponding to similar risk levels. Our analysis is built on a seminal idea by Galluccio, Bouchaud and Potters [3], who have restated the problem of finding solutions of the portfolio optimization problem in futures markets in terms of a spin glass problem. In order to get the best portfolio (i.e. the one lying on the efficiency frontier), we have to implement a twostep procedure. A worked example with real data is presented.
2003
 Optimization Strategies in Complex Systems
[Capitolo/Saggio]
L., Bussolari; P., Contucci; Giardina', Cristian; Giberti, Claudio; Unguendoli, Francesco; Vernia, Cecilia
abstract
We consider a class of combinatorial optimization problems that emerge in a variety of domains among which: condensed matter physics, theory of financial risks, error correcting codes in information transmissions, molecular and protein conformation, image restoration. We show the performances of two algorithms, the "greedy'' (quick decrease along the gradient) and the "reluctant'' (slow decrease close to the level curves) as well as those of a "stochastic convex interpolation'' of the two.Concepts like the average relaxation time and the wideness of theattraction basin are analyzed and their system size dependenceillustrated.
2003
 Thermodynamic Limit for MeanField Spin Models
[Articolo su rivista]
A., Bianchi; P., Contucci; Giardina', Cristian
abstract
If the BoltzmannGibbs state omega_N of a meanfield Nparticlesystem with Hamiltonian H_N verifies the condition omega_N(H_N) >=omega_N(H_{N_1}+H_{N_2}), for every decomposition N_1+N_2=N, then its freeenergy density increases with N. We prove such a condition for a wide class ofspin models which includes the CurieWeiss model, its pspin generalizations(for both even and odd p), its random field version and also the finite patternHopfield model. For all these cases the existence of the thermodynamic limit bysubadditivity and boundedness follows.
2003
 Thermodynamical limit for correlated Gaussian random energy models
[Articolo su rivista]
P., Contucci; M., Degli Esposti; Giardina', Cristian; S., Graffi
abstract
Let {E<sub>σ</sub> (N)}<sub>σ∈ΣN</sub> be a family of Σ<sub>N</sub> = 2<sup>N</sup> centered unit Gaussian random variables defined by the covariance matrix C<sub>N</sub> of elements c<sub>N</sub>(σ, τ): = Av(E<sub>σ</sub>(N)E<sub>τ</sub>(N)) and H<sub>N</sub>(σ) = √NE<sub>σ</sub>(N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N<sub>1</sub> + N<sub>2</sub>, and all pairs (σ, τ) ∈ Σ<sub>N</sub> × Σ<sub>N</sub>: c<sub>N</sub>(σ, τ) ≤ N<sub>1</sub>/N c<sub>N1</sub>(π<sub>1</sub>(τ), π<sub>1</sub>(τ)) + N<sub>2</sub>/N c<sub>N2</sub>(π<sub>2</sub>(σ), π<sub>2</sub>(τ)), where π<sub>k</sub> (τ), k = 1, 2 are the projections of σ ∈ Σ<sub>N</sub> into Σ<sub>Nk</sub>. The condition is explicitly verified for the SherringtonKirkpatrick, the even pspin, the Derrida REM and the DerridaGardner GREM models.
2002
 Portfolio optimization with shortselling and spinglass
[Articolo su rivista]
L., Bongini; Giardina', Cristian; M., Degli Esposti; A., Schianchi
abstract
In this paper, we solve a general problem of optimizing a portfolio in a futures markets framework, extending the previous work of Galluccio et al. [Physica A 259, 449 (1998)]. We allow for long buying/short selling of a relatively large number of assets, assuming a fixed level of margin requirement. Because of nonlinearity in the constraint, we derive a multiple equilibrium solution, in a size exponential respect to the number of assets. That means that we can not obtain the unique efficiency frontier, but many of them and each one is related to different levels of risk. Such a problem is analogous to that of finding the ground state in longranged Ising spin glass with external field. In order to get the best portfolio (i.e. that is along the best efficiency frontier), we have to implement a twostep procedure, performing the exhaustive enumeration of all local minima. We develop a concrete application, where the different part of the proposed solution are computed.
2002
 Screened Vortex Lattice Model with Disorder
[Working paper]
Giardina', Cristian; N. V., Priezjev; J. M., Kosterlitz
abstract
The three dimensional XY model with quenched random disorder and finite screening is studied. We argue that the system scales to model with $\lambda\simeq 0\simeq T$ and the resulting effective model is studied numerically by defect energy scaling. In zero external field we find that there exists a true superconducting phase with a stiffness exponent $\theta\simeq +1.0$ for weak disorder. For low magnetic field and weak disorder, there is also a superconducting phase with $\theta\simeq +1.0$ which we conjecture is a Bragg glass. For larger disorder or applied field, there is a non superconducting phase with $\theta\simeq 1.0$. We estimate the critical external field whose value is consistent with experiment.
2001
 Statistics of energy levels and zero temperature dynamics for deterministic spin models with glassy behaviour
[Articolo su rivista]
M., Degli Esposti; Giardina', Cristian; S., Graffi; S., Isola
abstract
We consider the zerotemperature dynamics for the infiniterange, non translation invariant onedimensional spin model introduced by Marinari, Parisi and Ritort to generate glassy behaviour out of a deterministic interaction. It is argued that there can be a large number of metastable (i.e., oneflip stable) states with very small overlap with the ground state but very close in energy to it, and that their total number increases exponentially with the size of the system.
2000
 Discrete spin variables and critical temperature in deterministic models with glassy behavior
[Articolo su rivista]
Giardina', Cristian
abstract
The problem of the existence of a glassy phase transition in deterministic spin models is reconsidered, examining an Ising model with general spin s and nontranslationally invariant interaction. The discrete nature of the spin variables is shown to allow the glass state.
2000
 Finite Thermal Conductivity in 1D Lattices
[Articolo su rivista]
Giardina', Cristian; R., Livi; A., Politi; M., Vassalli
abstract
We discuss the thermal conductivity of a chain of coupled rotators, showing that it is the first example of a ID nonlinear lattice exhibiting normal transport properties in the absence of an onsite potential. Numerical estimates obtained by simulating a chain in contact with two thermal baths at different temperatures are found to be consistent with those based on linear response theory. The dynamics of the Fourier modes provides direct evidence of energy diffusion. The finiteness of the conductivity is traced back to the occurrence of phase jumps. Our conclusions are confirmed by the analysis of two variants of this model.
1998
 Ergodic properties of microcanonical observables
[Articolo su rivista]
Giardina', Cristian; R., Livi
abstract
The problem of the existence of a strong stochasticity threshold in the FPUβ model is reconsidered, using suitable microcanonical observables of thermodynamic nature, like the temperature and the specific heat. Explicit expressions for these observables are obtained by exploiting rigorous methods of differential geometry. Measurements of the corresponding temporal autocorrelation functions locate the threshold at a finite value of the energy density, which is independent of the number of degrees of freedom.