Nuova ricerca Cristian GIARDINA' Professore Ordinario presso: Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Scienze Comunicazione

## Pubblicazioni

2020 - Due problemi di teoria analitica dei numeri: somme armoniche con i primi e distribuzione delle cifre di quozienti fra interi [Tesi di dottorato]
TONON, REMIS
abstract

In the first part of this thesis we extend the results of the paper “Small values of signed harmonic sums” by Bettin, Molteni & Sanna (2018). There, the authors consider harmonic truncated series, where the summands can have a positive or a negative sign; using these objects to approximate any real value, they study the function that measures the precision of this approximation. In particular, they prove some bounds for this function in some specific ranges. In this thesis, we prove that the same result holds not only for the sequence of all natural numbers, but also for any subsequence that satisfies a growth hypothesis. Besides, in the case of the sequence of numbers that are the product of k distinct primes, where k is a fixed natural number, we obtain a significant improvement on the bounds for the approximating function. In the second part of this thesis, we improve the result of the paper “Probability of digits by dividing random numbers: a ψ and ζ functions approach” by Gambini, Mingari Scarpello & Ritelli (2012). The authors study there the distribution of the nth digit after the decimal point (in different bases) of all possible ratios between the first N natural numbers: they prove that it is not uniform, but it follows a law analogous to Benford’s one. In this thesis, we improve the error term found by the authors; besides, we study some further and different aspects and some variations of the problem, such as the uniformity of the formula.

2020 - Esistenza di soluzioni per modelli semplificati di turbolenza. [Tesi di dottorato]
MONTAGNANI, ALESSANDRO
abstract

In the first two chapters of the thesis we show how can Shell Models be viewed as a simplified model of the Navier-Stokes equation. The Navier-Stokes equation, mimics the dynamic of a fluid and it represents one of the most challenging open problems in mathematic.In the NS model the fluid is thought as a continuum stream identified by a velocity field, a temperature field, a pressure field and a density field. From the Fourier series of the NSE we deduce a transfer of energy from large to small scale. Shell models are a simplified version of the Fourier series of NSE, with the aim of mimic the energy cascade in a infinite dimensional dynamic system where the equations are coupled, this means that the n-th component interacts only with n-1-th and n+1-th components. A shell model can be viewed as a division of the space into concentric spheres with expontially growing radius. In this environment the n-th shell will be the set of wave numbers contained in the n-th sphere and not in the n-1-th one. Compared to the NSE we can note that the dynamic of shell models is way simplier. Despite this, shell models are consistent enough with the turbulence theory to significant mimic the energy cascade of NSE. The aim of the third chapter is to prove an existence result on the mixed shell model extending the classic standard existence results from finite energy initial conditions to M-almost every initial conditions, where M is a Gaussian measure on the infinite dimensional space of initial conditions. The first step would be to consider a mixed model with suitable coefficients, to let to a certain Gaussian measure M to be invariant for the system. Then we find a Galerkin approximation of the infinite dimensional shell model, in a way that every finite dimensional system of the sequence admits a unique solution by Cauchy-Lipschitz theorem and such that every finite N-dimensional system has an invariant measure given by the projection of M on the first N coordinates. In the second step we introduce random initial conditions and we obtain fundamental extimates on the norm of random solutions on certain spaces. The invariance of the measure plays an important role in the computation of the norm extimates and our results on these extimates fully depend on it. In the third step we use a compactness argument to extract a weak limit of the sequence of random solution for the N-dimensional system, based on a combination of Aubin-Lions lemma and Prohorov Theorem. Last, in the fourth step, we prove that the weak limit obtained from the third step can be extended to an a.s. limit, thanks to Skorokhod representation theorem, that formally solves the integral equation of the infinite dimensional dynamic system. In the fourth and last chapter we work on tree models. First we briefly introduce turbolence tree models, then we work on a forced tree model, way more general than the Katz-Pavlovic one, where the force acts only on the first component. Differently from other shell models, this forced ones is not conservative, in the sense that the energy is not constant along the trajectories. Last we consider a mixed cascade tree model with coefficients taken to let to a Gaussian measure to be invariant and we apply all techniques used in the last chapter to get an existence result that improves the finite energy existence result that we have done for the more general model.

2020 - I Metodi Interior Point Incontrano le Reti Neurali: un'Applicazione al Deblurring di Immagini [Tesi di dottorato]
BERTOCCHI, CARLA
abstract

The aim of this thesis is to propose novel Deep Learning model to approach the image deblurring problem. This is a well known Inverse Problem that is usually reformulated as a regularized optimization problem, in which the objective function to be minimized is a sum of a data discrepancy measure with a regularization term. Also additional constraints can be imposed to incorporate a priori knowledge on the desired solution. In our work we consider smooth data-fidelity and regularization terms and we include constraints in the objective function by means of a logarithmic barrier. A proximal interior point method (IPM) is adopted to address the minimization step, in which the proximity operator is restricted only to the barrier function. The key issue of our proposed approach is the following: the regularization parameter, the barrier parameter and the step size needed in the iterations of the IPM are chosen by means of a particular Deep Learning strategy. In particular, we say that the IPM algorithm is unfolded in a neural network structure, whose training process merges with the optimization process. We used benchmarks image datasets to train the resulting neural network architecture and test our approach. Comparisons with standard gradient projection methods, with recent machine learning algorithms and also with other unfolded methods have been performed and the tests showed good performances and competitiveness of our approach.

2020 - La previsione geometrica: un modello per analizzare un processo cognitivo inerente il problem-solving in geometria [Tesi di dottorato]
MIRAGLIOTTA, ELISA
abstract

The purpose of the research is to study cognitive aspects of how geometric predictions are produced during problem-solving activities in Euclidean geometry. The process of geometric prediction is seen as a specific visuo-spatial ability involved in geometrical reasoning. Indeed, when solvers engage in solving a geometrical problem, they can imagine the consequences of transformations of the figure; such transformations can be more or less coherent with the theoretical constraints given by the problem, and the products of such transformations can hinder or promote the problem-solving process. Previous research has stressed the dual nature of geometrical objects, intertwining a conceptual component and a figural component. Interpreting geometrical reasoning in terms of a dialectic between these two aspects (Fischbein, 1993), this study aims at gaining insight into the cognitive process of geometric prediction, a process through which a figure is manipulated, and its change is imagined, while certain properties are maintained invariant. This process is described through a model of prediction-generation elaborated cyclically by observing, analyzing through a microgenetic approach, and re-analyzing solvers’ resolution of prediction open problems in a paper-and-pencil environment and in a Dynamic Geometry Environment (DGE). The prediction open problems designed were proposed during task-based interviews to participants selected on a voluntary basis. Participants were a total of 37 Italian high school students and undergraduate, graduate and PhD students in mathematics. Data are composed of video and audio recordings, transcriptions, solvers’ drawings. The final version of the model provides a description of the prediction processes accomplished by a solver who engages in the resolution of prediction open problems proposed in this study; it provides a lens through which solvers’ productions can be analyzed and it provides insight into prediction processes. In particular, it sheds light onto the key role played by theoretical elements that are introduced by the solvers during the resolution process and the key role played by the solver’s theoretical control. The study has implications for the design of activities, especially at the high school level, with the educational objective of fostering students’ geometrical reasoning and in particular their theoretical control over the geometrical figures.

2020 - Proprietà spettrali dei metodi del gradiente per problemi di ottimizzazione con vincoli speciali [Tesi di dottorato]
CRISCI, SERENA
abstract

2020 - Simulazioni Numeriche di Opportune Equazioni dell'Elettromagnetismo Applicate al Caso di un'Antenna Biconica [Tesi di dottorato]
DIAZZI, LORENZO
abstract

By working with an extensions of the classical set of electromagnetic equations, we implemented some numerical techniques to study the near-field of a biconic antenna. Though the usual Maxwell's equations are included in the model, the generalization is necessary to handle the possible creation of regions displaying non-vanishing divergence in proximity of the boundaries, where perfect conductivity is not given for granted. Finite-difference schemes have been primarily used in a three-dimensional domain described by cylindrical coordinates. The numerical experiments include the simulation of solitary waves in vacuum and their behaviour when passing through media of different conductivity. In a successive development these waves are studied in conjunction with boundary constraints, due to the their interaction with the conductive guides. The goal of this analysis, only in part achieved, is a full understanding of the passage of the electromagnetic wave from the state of guided evolution to the one when the signal travels in free space.

2020 - Strutture di incidenza finite con blocchi di forma assegnata [Tesi di dottorato]
FERNICOLA, FRANCO
abstract

In this thesis we consider the problem of decomposing the complete graph on v vertices into subgraphs, all of which are isomorphic to a given graph H. The subgraphs of the decomposition are called blocks. According to the definition of a decomposition each edge of the complete graph must occur in precisely one blocjk of the decomposition. This notion generalizes the idea of a block design. From this point of view a block design is a decomposition of the complete graph into complete subgraphs, all having equal cardinality k. In the definition of a block design blocks have no additional structure other than that of bare subsets of the set of points. Therefore, the notion of an H-design which is studied in this thesis forces blocks to inherit a certain structure, namely that of being a copy of the graph H, which, in some sense, determines the “shape” of the blocks. One of the main problems in the theory of classical block designs as well as in the theory of H-designs consists in the determination of the existence spectrum, that is the determination of the values v for which the block design exists. Open problems related to the existence spectrum do survive, as it is known, even for incidence structures that were studied long before block designs, say, for instance, finite projective planes. In the case of interest for this thesis, the request is to establish, for a given graph H, the existence spectrum for H-designs, that is the set of all values v for which an H-deecomposition of the complete graph on v vertices exists. A contribution to the determination of the spectrum is obtained in case H is a connected graph with 7 vertices and 7 edges containing a cycle of length 3. This case remains open from previous investigations in which the graph H is smaller or contains a longer cycle. Generalizations in various directions are also approached, assuming a slight change in the type of the graph H or assuming special incidence features for the decomposition, such as those that generally involve some kind of balance, typically the request that a certain parameter of the decomposition remain uniform consideration. In our specific context the decomposition is balanced if the number of blocks containing any given vertex is constant.

2020 - Sulle trasformazioni cremoniane piane di grado basso e le loro lunghezze quadratiche [Tesi di dottorato]
NGUYEN, THI NGOC GIAO
abstract

2020 - Un framework per l’analisi dei sistemi di apprendimento automatico [Tesi di dottorato]
SPALLANZANI, MATTEO
abstract

Making predictions is about getting insights into the patterns of our environment. We can access the physical world through media, measuring instruments, which provide us with data in which we hope to find useful patterns. The development of computing machines has allowed storing large data sets and processing them at high speed. Machine learning studies systems which can automate the detection of patterns in large data sets using computers. Machine learning lies at the core of data science and artificial intelligence, two research fields which are changing the economy and the society in which we live. Machine learning systems are usually trained and deployed on powerful computer clusters composed by hundreds or thousands of machines. Nowadays, the miniaturisation of computing devices is allowing deploying them on battery-powered systems embedded into diverse environments. With respect to computer clusters, these devices are far less powerful, but have the advantage of being nearer to the source of the data. On one side, this increases the number of applications of machine learning systems; on the other side, the physical limitations of the computing machines require identifying proper metrics to assess the fitness of different machine learning systems in a given context. In particular, these systems should be evaluated according not only to their modelling and statistical properties, but also to their algorithmic costs and their fitness to different computer architectures. In this thesis, we analyse modelling, algorithmic and architectural properties of different machine learning systems. We present the fingerprint method, a system which was developed to solve a business intelligence problem where statistical accuracy was more important than latency or energy constraints. Then, we analyse artificial neural networks and discuss their appealing computational properties; we also describe an example application, a model we designed to identify the objective causes of subjective driving perceptions. Finally, we describe and analyse quantized neural networks, artificial neural networks which use finite sets for the parameters and step activation functions. These limitations pose challenging mathematical problems, but quantized neural networks can be executed extremely efficiently on dedicated hardware accelerators, making them ideal candidates to deploy machine learning on edge computers. In particular, we show that quantized neural networks are equivalent to classical artificial neural networks (at least on the set of targets represented by continuous functions defined on compact domains); we also present a novel gradient-based learning algorithm for, named additive noise annealing, based on the regularisation effect of additive noise on the argument of discontinuous functions, reporting state-of-the-art results on image classification benchmarks.

2019 - Non-compact 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 one-dimensional 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 non-equilibrium 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 imaginary-time. 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 (2|1) superstring that has been derived directly from =4 SYM.

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 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.

2019 - Stationary States in Infinite Volume with Non-zero 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 non-zero current in three or more dimensions.

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 two-dimensional 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 two-dimensional 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 - Self-Duality 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 self-duality 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 self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual 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 self-duality 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 time-scale for the condensate motion. In this case we compute this typical time-scale 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 time-scales emerges. We prove such a scenario, involving two additional time-scales, in a one-dimensional setting with two metastable states and nearest-neighbor jumps.

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 rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular 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 $0\le \beta^{\mathrm{an}}_c < \infty$ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature $\beta$ and external magnetic field $B$ for which either $\beta<\beta^{\mathrm{an}}_c$ and $B=0$, or $\beta>0$ and $B\neq 0$. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters $\beta$ and $B$, because phase transitions do not exist for these systems as they are closely related to one-dimensional 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 Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional 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 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
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 Curie-Weiss 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 Curie-Weiss 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 rank-1 inhomogeneous Curie-Weiss 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 non-classical 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) Curie-Weiss 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(τ-2)/(τ-1) converges in law to some limiting random variable whose distribution we explicitly characterize.

2015 - 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 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.

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 , 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.

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 tree-like 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 2-regular 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 non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, 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 time-reversal 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 so-called locally tree-like random graphs. We pay special attention to trees and graphs with a power-law offspring or degree distribution whose tail behavior is characterized by its power-law 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.  and Leone et al. . These values depend on the power-law exponent , taking the mean-field values for > 5, but different values for ∈ (3, 5).

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

This paper is a follow-up of the work initiated in , 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 Erdős-Rényi Random Graph [Articolo su rivista]
Pierluigi, Contucci; Sander, Dommers; Giardina', Cristian; Shannon, Starr
abstract

We study the antiferromagnetic Potts model on the Poissonian Erdős-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment 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 Derrida-Ruelle 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 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 - Interaction Flip Identities for non Centered Spin Glasses [Articolo su rivista]
P., Contucci; Giardina', Cristian; Giberti, Claudio
abstract

We consider spin glass models with non-centered 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 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.

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 up-to-date 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 spin-glass 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, 765-783, (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 Edwards-Anderson 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 well-known 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 so-called 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 power-law 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 power-law 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 tree-like 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

Real-world 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 Kipnis-Marchioro-Presutti (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 - Interaction-Flip 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 non-interacting random field is analyzed and the triviality of its overlap distribution proved.

2009 - Matching with shift for one-dimensional 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 Edward-Anderson 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 quantum-like 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 Edwards-Anderson 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 Edwards-Anderson Model" PRL 99, 057206 (2007). We show that the procedure developed in the aforementioned paper to detect ultrametricity is able to discriminate the non-ultrametric behavior of the two-dimensional Edwards-Anderson model from the ultrametric three-dimensional one. Moreover, the interesting finding of Jorg and Krzakala that in the two-dimensional Edwards-Anderson 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 long-range 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 Ghirlanda-Guerra identities. [Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract

If the variance of a Gaussian spin-glass Hamiltonian grows like the volume the model fulfills the Ghirlanda-Guerra identities in terms of the normalized Hamiltonian covariance. © Springer Science+Business Media, LLC 2007.

2007 - Ultrametricity in the Edwards-Anderson model. [Articolo su rivista]
P., Contucci; GIARDINA', Cristian; GIBERTI, Claudio; G., Parisi; VERNIA, Cecilia
abstract

We test the property of ultrametricity for the spin-glass three-dimensional Edwards-Anderson 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 3-dimensional Ising spin glasses'', cond-mat/0608535v2 [Working paper]
P., Contucci; Giardina', Cristian
abstract

We comment on recent numerical experiments by G.Hed and E.Domany [cond-mat/0608535v2] on the quenched equilibrium state of the Edwards-Anderson 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 cond-mat/0608535v4

2006 - Direct evaluation of large-deviation 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 large-deviation 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 Edwards-Anderson 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 three-dimensional Gaussian Edwards-Anderson 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 "trivial-nontrivial" picture should be dismissed. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model. © 2006 The American Physical Society.

2006 - Relative entropy and waiting times for continuous-time Markov processes [Articolo su rivista]
J. R., Chazottes; Giardina', Cristian; F., Redig
abstract

For discrete-time 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 continuous-time setting. Contrarily to the discrete-time 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 continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of the logarithm of waiting-times ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle).

2005 - Factorization properties in the three-dimensional Edwards-Anderson model [Articolo su rivista]
P., Contucci; Giardina', Cristian
abstract

We study the three-dimensional Gaussian Edwards-Anderson model and find numerical evidence of a simple factorization law of the link-overlaps 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 factorization-triviality 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 greedy-reluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the Sherrington-Kirkpatrick model of spin-glasses. © 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 NP-complete 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 - Spin-glass 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 β-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson 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 momentum-conserving 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 vortex-vortex 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 d-dimensional 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 Marinari-Parisi-Ritort [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 Sherrington-Kirkpatrick model. Here we compute the energy distribution, and work out an estimate for the two-point correlation function. Moreover, we show an exponential increase with the system size of the number of metastable states also for non-zero magnetic field.

2003 - Energy-decreasing dynamics in mean-field spin models [Articolo su rivista]
L., Bussolari; P., Contucci; M., Degli Esposti; Giardina', Cristian
abstract

We perform a statistical analysis of deterministic energy-decreasing algorithms on mean-field spin models with a complex energy landscape, such as the Sine model and the Sherrington-Kirkpatrick model. We specifically address the following question: in the search for low-energy 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 short-selling [Articolo su rivista]
L., Bongini; M., Degli Esposti; Giardina', Cristian; A., Schianchi
abstract

In this paper an extension of the Lintner model  is considered: the problem of portfolio optimization is studied when short-selling is allowed through the mechanism of margin requirements. This induces a non-linear 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 short-selling. This leads to a fully non-linear 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 , who have re-stated 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 two-step 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 Mean-Field Spin Models [Articolo su rivista]
A., Bianchi; P., Contucci; Giardina', Cristian
abstract

If the Boltzmann-Gibbs state omega_N of a mean-field N-particlesystem 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 Curie-Weiss model, its p-spin 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 Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.

2002 - Portfolio optimization with short-selling and spin-glass. [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 non-linearity 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 long-ranged 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 two-step 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 zero-temperature dynamics for the infinite-range, non translation invariant one-dimensional 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., one-flip 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 on-site 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 FPU-β 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.