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The presence of a random magnetic field in ferromagnetic systems leads, in the broken phase, to an anomalous $O(\sqrt{1/N})$ convergence of some thermodynamic quantities to their asymptotic limits. Here we show a general method, based on the replica trick, to compute analytically the $O(\sqrt{1/N})$ finite size correction to the average free energy. We apply this method to two mean field Ising models, fully connected and random regular graphs, and compare the results to exact numerical algorithms. We argue that this behaviour is present in finite dimensional models as well.

We propose a new algorithm for inferring the state of hidden spins and reconstructing the connections in a synchronous kinetic Ising model, given the observed history. Focusing on the case in which the hidden spins are conditionally independent of each other given the state of observable spins, we show that calculating the likelihood of the data can be simplified by introducing a set of replicated auxiliary spins. Belief propagation (BP) and susceptibility propagation (SusP) can then be used to infer the states of hidden variables and to learn the couplings. We study the convergence and performance of this algorithm for networks with both Gaussian-distributed and binary bonds. We also study how the algorithm behaves as the fraction of hidden nodes and the amount of data are changed, showing that it outperforms the Thouless-Anderson-Palmer (TAP) equations for reconstructing the connections. QC 20150625

In many mean-field glassy systems, the low-temperature Gibbs measure is dominated by exponentially many metastable states. We analyze the evolution of the metastable states as temperature changes adiabatically in the solvable case of the spherical $s+p$-spin glass model, extending the work of Barrat, Franz and Parisi J. Phys. A 30, 5593 (1997). We confirm the presence of level crossings, bifurcations, and temperature chaos. For the states that are at equilibrium close to the so-called dynamical temperature $T_d$, we find, however, that the following state method (and the dynamical solution of the model as well) is intrinsically limited by the vanishing of solutions with non-zero overlap at low temperature.

We uncover a new kind of entropic long range order in finite dimensional spin glasses. We study the link-diluted version of the Edwards-Anderson spin glass model with bimodal couplings (J=+/-1) on a 3D lattice. By using exact reduction algorithms, we prove that there exists a region of the phase diagram (at zero temperature and link density low enough), where spins are long range correlated, even if the ground states energy stiffness is null. In other words, in this region twisting the boundary conditions cost no energy, but spins are long range correlated by means of pure entropic effects., Comment: 15 pages, 6 figures. v3: added a phase diagram for ferromagnetically biased couplings

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved., 30 pages, 4 figures

We investigate the evolution of a population of non-interacting particles which undergo diffusion and multiplication. Diffusion is assumed to be homogeneous, while multiplication proceeds with different rates reflecting the distribution of nutrients. We focus on the situation where the distribution of nutrients is a stationary quenched random variable, and show that the population exhibits a super-exponential growth whenever the nutrient distribution is unbounded. We elucidate a huge difference between the average and typical asymptotic growths and emphasize the role played by the spatial correlations in the nutrient distribution., 15 pages, 1 figure

We present a massive equilibrium simulation of the three-dimensional Ising spin glass at low temperatures. The Janus special-purpose computer has allowed us to equilibrate, using parallel tempering, L=32 lattices down to T=0.64 Tc. We demonstrate the relevance of equilibrium finite-size simulations to understand experimental non-equilibrium spin glasses in the thermodynamical limit by establishing a time-length dictionary. We conclude that non-equilibrium experiments performed on a time scale of one hour can be matched with equilibrium results on L=110 lattices. A detailed investigation of the probability distribution functions of the spin and link overlap, as well as of their correlation functions, shows that Replica Symmetry Breaking is the appropriate theoretical framework for the physically relevant length scales. Besides, we improve over existing methodologies to ensure equilibration in parallel tempering simulations., 48 pages, 19 postscript figures, 9 tables. Version accepted for publication in the Journal of Statistical Mechanics

An efficient microcanonical dynamics has been recently introduced for Ising spin models embedded in a generic connected graph even in the presence of disorder i.e. with the spin couplings chosen from a random distribution. Such a dynamics allows a coherent definition of local temperatures also when open boundaries are coupled to thermostats, imposing an energy flow. Within this framework, here we introduce a consistent definition for local energy currents and we study their dependence on the disorder. In the linear response regime, when the global gradient between thermostats is small, we also define local conductivities following a Fourier dicretized picture. Then, we work out a linearized "mean-field approximation", where local conductivities are supposed to depend on local couplings and temperatures only. We compare the approximated currents with the exact results of the nonlinear system, showing the reliability range of the mean-field approach, which proves very good at high temperatures and not so efficient in the critical region. In the numerical studies we focus on the disordered cylinder but our results could be extended to an arbitrary, disordered spin model on a generic discrete structures., Comment: 12 pages, 6 figures

We consider the Ising model for two interacting groups of spins embedded in an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are investigated by means of extensive Monte Carlo simulations. Our results evidence the existence of a phase transition at a value of the inter-groups interaction coupling $J_{12}^C$ which depends algebraically on the dilution of the graph and on the relative width of the two populations, as explained by means of scaling arguments. We also measure the critical exponents, which are consistent with those of the Curie-Weiss model, hence suggesting a wide robustness of the universality class., Comment: 11 pages, 4 figures

The aim of this work is to try to bridge over theoretical immunology and disordered statistical mechanics. Our long term hope is to contribute to the development of a quantitative theoretical immunology from which practical applications may stem. In order to make theoretical immunology appealing to the statistical physicist audience we are going to work out a research article which, from one side, may hopefully act as a benchmark for future improvements and developments, from the other side, it is written in a very pedagogical way both from a theoretical physics viewpoint as well as from the theoretical immunology one. Furthermore, we have chosen to test our model describing a wide range of features of the adaptive immune response in only a paper: this has been necessary in order to emphasize the benefit available when using disordered statistical mechanics as a tool for the investigation. However, as a consequence, each section is not at all exhaustive and would deserve deep investigation: for the sake of completeness, we restricted details in the analysis of each feature with the aim of introducing a self-consistent model., Comment: 22 pages, 14 figure

We introduce a new microcanonical dynamics for a large class of Ising systems isolated or maintained out of equilibrium by contact with thermostats at different temperatures. Such a dynamics is very general and can be used in a wide range of situations, including disordered and topologically inhomogenous systems. Focusing on the two-dimensional ferromagnetic case, we show that the equilibrium temperature is naturally defined, and it can be consistently extended as a local temperature when far from equilibrium. This holds for homogeneous as well as for disordered systems. In particular, we will consider a system characterized by ferromagnetic random couplings $J_{ij} \in [ 1 - \epsilon, 1 + \epsilon ]$. We show that the dynamics relaxes to steady states, and that heat transport can be described on the average by means of a Fourier equation. The presence of disorder reduces the conductivity, the effect being especially appreciable for low temperatures. We finally discuss a possible singular behaviour arising for small disorder, i.e. in the limit $\epsilon \to 0$., Comment: 14 pages, 8 figures

Domain walls, optimal droplets and disorder chaos at zero temperature are studied numerically for the solid-on-solid model on a random substrate. It is shown that the ensemble of random curves represented by the domain walls obeys Schramm's left passage formula with kappa=4 whereas their fractal dimension is d_s=1.25, and therefore is NOT described by "Stochastic-Loewner-Evolution" (SLE). Optimal droplets with a lateral size between L and 2L have the same fractal dimension as domain walls but an energy that saturates at a value of order O(1) for L->infinity such that arbitrarily large excitations exist which cost only a small amount of energy. Finally it is demonstrated that the sensitivity of the ground state to small changes of order delta in the disorder is subtle: beyond a cross-over length scale L_delta ~ 1/delta the correlations of the perturbed ground state with the unperturbed ground state, rescaled by the roughness, are suppressed and approach zero logarithmically., 23 pages, 11 figures

We provide an exact expression of the moment of the partition function for random energy models of finite system size, generalizing an earlier expression for a grand canonical version of the discrete random energy model presented by the authors in Prog. Theor. Phys. 111, 661 (2004). The expression can be handled both analytically and numerically, which is useful for examining how the analyticity of the moment with respect to the replica numbers, which play the role of powers of the moment, can be broken in the thermodynamic limit. A comparison with a replica method analysis indicates that the analyticity breaking can be regarded as the origin of the one-step replica symmetry breaking. The validity of the expression is also confirmed by numerical methods for finite systems., 16 pages, 4 figures

For a non-interacting Bose gas on a lattice we compute the shift of the critical temperature for condensation when random-bond and onsite disorder are present. We evidence that the shift depends on the space dimensionality D and the filling fraction f. For D -> infinity (infinite-range model), using results from the theory of random matrices, we show that the shift of the critical temperature is negative, depends on f, and vanishes only for large f. The connections with analogous results obtained for the spherical model are discussed. For D=3 we find that, for large f, the critical temperature Tc is enhanced by disorder and that the relative shift does not sensibly depend on f; at variance, for small f, Tc decreases in agreement with the results obtained for a Bose gas in the continuum. We also provide numerical estimates for the shift of the critical temperature due to disorder induced on a non-interacting Bose gas by a bichromatic incommensurate potential., 18 pages, 8 figures; Fig. 8 improved adding results for another value of q (q=830/1076)

We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the +- J Ising model along the paramagnetic-ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size-dependence of suitably defined autocorrelation times at the critical point provides the estimate z=2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from T=\infty to T=T_c. In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behavior gives an estimate of z that is perfectly consistent with the equilibrium estimate z=2.35(2)., 38 pages

We investigate a disordered version of a thermodynamic fiber bundle model proposed by Selinger, Wang, Gelbart, and Ben-Shaul a few years ago. For simple forms of disorder, the model is analytically tractable and displays some new features. At either constant stress or constant strain, there is a non monotonic increase of the fraction of broken fibers as a function of temperature. Moreover, the same values of some macroscopic quantities as stress and strain may correspond to different microscopic cofigurations, which can be essential for determining the thermal activation time of the fracture. We argue that different microscopic states may be characterized by an experimentally accessible analog of the Edwards-Anderson parameter. At zero temperature, we recover the behavior of the irreversible fiber bundle model., 18 pages, 10 figures

We apply path-integral techniques to study the dynamics of agent-based models with asymmetric information structures. In particular, we devise a batch version of a model proposed originally by Berg et al. [Quant. Fin. 1 (2001) 203], and convert the coupled multi-agent processes into an effective-agent problem from which the dynamical order parameters in ergodic regimes can be derived self-consistently together with the corresponding phase structure. Our dynamical study complements and extends the available static theory. Results are confirmed by numerical simulations., minor revision of text, accepted by JSTAT

Crystalline plasticity is strongly interlinked with dislocation mechanics and nowadays is relatively well understood. Concepts and physical models of plastic deformation in amorphous materials on the other hand - where the concept of linear lattice defects is not applicable - still are lagging behind. We introduce an eigenstrain-based finite element lattice model for simulations of shear band formation and strain avalanches. Our model allows us to study the influence of surfaces and finite size effects on the statistics of avalanches. We find that even with relatively complex loading conditions and open boundary conditions, critical exponents describing avalanche statistics are unchanged, which validates the use of simpler scalar lattice-based models to study these phenomena., Comment: Journal of Statistical Mechanics: Theory and Experiment, 2015, P02011

We present a method derived by cavity arguments to compute the spin-glass and higher-order susceptibilities in diluted mean-field spin-glass models. The divergence of the spin-glass susceptibility is associated to the existence of a non-zero solution of a homogeneous linear integral equation. Higher order susceptibilities, relevant for critical dynamics through the parameter exponent $\lambda$, can be expressed at criticality as integrals involving the critical eigenvector. The numerical evaluation of the corresponding analytic expressions is discussed. The method is illustrated in the context of the de Almeida-Thouless line for a spin-glass on a Bethe lattice but can be generalized straightforwardly to more complex situations., Comment: v3: typo error in eq. (10) (and similar equations) corrected

We study the performance of different message passing algorithms in the two dimensional Edwards Anderson model. We show that the standard Belief Propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a Generalized Belief Propagation (GBP) algorithm, derived from a Cluster Variational Method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: Double Loop (DL) and a two-ways message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to non paramagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude., Comment: 19 pages, 13 figures

In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More specificly, the average of the overlap, which is naively expected to take values in the whole interval $[-1,+1]$, becomes positive if we ``first'' apply an external field, so to destroy the gauge invariance of the model, and ``then'' remove it in the thermodynamic limit. This phenomenon emerges at the critical point. This first result is weaker that the one obtained by Talagrand (not limited to the average of the overlap), but we show here that, at least in average, the overlap is proven to be non-negative with no use of the Ghirlanda-Guerra identities. The latter are instead needed to obtain the second result, which is the control the behavior of the overlap at the critical point: we find the critical exponents of all the overlap correlation functions., Comment: 19 pages