There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Introduction to quality engineering : designing quality into products and processes

Fluctuation-dissipation: Response theory in statistical physics

1 Mean-field theory of phase transitions 2 Mean-field theory of spin glasses 3 Replica symmetry breaking 4 Gauge theory of spin glasses 5 Error-correcting codes 6 Image restoration 7 Associative memory 8 Learning in perceptron 9 Optimization problems A Eigenvalues of the Hessian B Parisi equation C Channel coding theorem D Distribution and free energy of K-Sat References Index

Statistical physics of spin glasses and information processing : an introduction

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. Long-range 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.

/pdf/duality-for-stochastic-models-of-transport-65yuvf3s5w.pdf

Duality for Stochastic Models of Transport

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 particles 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. Long-range 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.

Duality for stochastic models of transport

http://www.ccbss.unimore.it/giardina/docs/1302.3206v2.pdf

Dualities in population genetics: a fresh look with new dualities.

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.

Nonequilibrium two-dimensional Ising model with stationary uphill diffusion.

We test the property of ultrametricity for the spin-glass three-dimensional Edwards-Anderson model in zero magnetic field with numerical simulations up to 20 3 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. Ultrametricity is a widely accepted property of the mean field spin-glass theory: it is a crucial ingredient in the field theoretical computations of the Sherrington-Kirkpatrick model [1‐3] as well as a guiding principle for the rigorous proof of its free energy density formula [4,5]. Its relevance in finite dimensional systems is nonetheless still an open matter, subject of intense investigations and debates in the theoretical and mathematical physics communities. Ultrametricity states a very striking property for a physical system: essentially it says that the equilibrium configurations of a large system can be classified in a taxonomic (hierarchical) way (as animal in different taxa): configurations are grouped in states, states are grouped in families, families are grouped in superfamilies. This equilibrium ultrametricity has a correspondence in the existence of widely separated time scales in the dynamics, typically of a glassy system. It is not clear at the present moment if ultrametricity is present in three-dimensional systems; the most studied case is three-dimensional spin glasses where contrasting results have been presented in the literature in the past 20 years. Part of the difficulties arise from the fact that ultrametricity should be, at the best, exact when the volume of the system goes to infinity and therefore simulations done on a limited range of volume are difficult to interpret. In this Letter we study systems ranging from 4 3 to 20 3 extending of about an order of magnitude the range of volume used in previous simulations. From the technical point of view ultrametricity implies that sampling three configurations independently with respect to their common Boltzmann-Gibbs state and averaging over the disorder, the distribution of the distances among them is supported, in the limit of very large systems, only on equilateral and isosceles triangles with no scalene triangles contribution. In a generic situation the relative weight of equilateral and isosceles triangles is arbitrary, however, it is well established in stochastically stable systems; the stochastic stability property was introduced for the infinite-range spin-glass model in [6,7] and later proved also for the realistic short-ranged models in finite dimensions [8,9]. The property of ultrametricity and the nontrivial structure of the overlap distribution are the characterizing features of the mean field picture and are mutually intertwined: a trivial (� -like) overlap probability distribution, like the one predicted in the droplet theory [10], is not compatible in fact with the previous ultrametric structure because it predicts only equilateral triangles all of the same side. In this Letter we study the Edwards-Anderson model [11] for the spin glass in the three-dimensional cubic lattice ~

/pdf/ultrametricity-in-the-edwards-anderson-model-1n9a5nbh4q.pdf

Claudio Giberti

Papers

Duality for Stochastic Models of Transport

Duality for stochastic models of transport

Dualities in population genetics: a fresh look with new dualities.

Nonequilibrium two-dimensional Ising model with stationary uphill diffusion.

Ultrametricity in the Edwards-Anderson model.