We study the spin system with long-range interactions and show the exactness of the mean-field theory under certain mild conditions [1]. This is a generalization of the result of Mori [2] for the non-random and spin-glass cases. To treat random fields, we evoke the self-averaging property of a function of random fields, which is our new contribution to the framework established in [2]. The result is that the mean-field theory gives the exact expression of the canonical free energy for systems with power-decaying interactions if the power is smaller than the spatial dimension. This is a highly non-trivial result because each site is given a very different environment by site-dependent random fields whereas the mean-field theory highlights the averages of physical quantities over many sites.

[1] J. Phys. Soc. Jpn. (to be published)
[2] T. Mori, "Instability of the mean-field states and generalization of phase separation in long range interacting systems", Phys. Rev. E 84, 031128 (2011)

The problem of error correction for Gallager's low-density parity-check codes is notoriously equivalent to that of computing marginal Boltzmann probabilities for an Ising-like model with multispin interactions in a nonuniform magnetic field. Since the graph of interactions is locally a tree, the solution is very well approximated by a generalized mean-field (Bethe-Peierls) approximation. Belief propagation (BP) and similar iterative algorithms are an efficient way to perform the calculation, but they sometimes fail to converge, giving rise to a nonnegligible residual error probability (error floor). On the other hand, provably-convergent algorithms are far too complex to be implemented in a real decoder. In this work we consider the application of the probability damping (PD)* technique, which can be regarded either as a variant of BP, of which it retains the low complexity, or as an approximation of a provably-convergent algorithm, from which it is expected to inherit better convergence properties. We investigate the algorithm behavior on a real instance of Gallager code, and compare the results with state-of-the-art algorithms.

* Please note that such an acronym was coined well before the foundation of the italian Partito Democratico.

I will present results on the effects of long-range couplings in two lattice models: (i) O\((N)\) models; and (ii) non-interacting fermions. For (i) we study O\((N)\) models with power-law interactions by using functional renormalization group techniques: we show that both in the Local Potential Approximation (LPA) and in LPA' all the universality classes are the same of the ones of the corresponding short-range O\((N)\) model at an effective fractional dimension, so that the critical exponents can be computed for the short-range ones. For (ii) we study the effects on the entanglement entropy of the long-range-ness of the hoppings for a model of free fermions on a lattice, pointing out that deviations from the area law can emerge in presence of a magnetic phase or randomness.

It has recently been discovered that single neuron stimulation can impact network dynamics in immature and adult neuronal circuits. Here we report a novel mechanism which can explain in developing neuronal circuits, typically composed of only excitatory cells, the peculiar role played by a few specific neurons in promoting/arresting the population activity. For this purpose, we consider a standard neuronal network model, with short-term synaptic plasticity,whose population activity is characterized by bursting behavior. The addition of developmentally regulated constraints on single neuron excitability and connectivity leads to the emergence of functional hub neurons, whose perturbation (through stimulation or deletion) is critical for the network activity. Functional hubs form a clique, where a precise sequential activation of the neurons is essential to ignite collective events without any need for a specific topological architecture.

We discuss a heterogeneous mean field approach to neural dynamics on random networks, that explicitly preserves the disorder in the structure of connections and leads to a set of self consistent equations. Within this approach, we provide an effective description of microscopic and large scale temporal signals in a leaky integrate and fire model with short term plasticity. Furthermore, we formulate and solve a global inverse problem of reconstructing the network in-degree distribution from the knowledge of the average activity field. The method is very general and applies to a large class of dynamical models on random networks.

I consider the \(\alpha\)-FPU system with \(N = 16,32\) and \(64\) masses connected by a nonlinear quadratic spring. The approach is based on resonant wave-wave interaction theory. I will show that the route to thermalization consists of three stages. The first one is associated with non-resonant three-wave interactions. At this short time scale, the dynamics is reversible; this stage coincides with the observation of recurrent phenomena in numerical simulations of the \(\alpha\)-FPU. On a larger time scale, exact four-wave resonant interactions start to take place; however, all quartets are isolated, preventing a full mixing of energy in the spectrum and thermalization. The last stage corresponds to six-wave resonant interactions. Those are responsible for the energy equipartition recently observed in numerical simulations. A key role in the finding is played by the Umklapp resonant interactions, typical of discrete systems.

The results are obtained in collaboration with Y. Lvov, D. Proment and L. Vozella.

A walker moves on a two dimensional square lattice, the landscape. At every site of the lattice there is an obstacle that determines the walker's steps. The obstacle has two possible orientations, say left and right, and the walker alters the landscape by changing the orientation of the obstacle as he passes. There are two types of obstacles, mirrors, or rotors [1]. Starting with a fraction \(p\) of right obstacles chosen at random we show that for some values of \(p\) there can be abnormal diffusion.

1 H. F. Meng, E. G. D. Cohen, Phys. Rev. E 50 2482 (1994); X. P. Kong, E. G. D.Cohen, J. Stat. Phys. 62 1153 (1991).

Several neurological disorders are associated with the aggregation of aberrant proteins, often localized in intracellular organelles such as the endoplasmic reticulum. Here we study protein aggregation kinetics by mean-field reactions and three dimensional Monte carlo simulations of diffusion-limited aggregation of linear polymers in a confined space, representing the endoplasmic reticulum. By tuning the rates of protein production and degradation, we show that the system undergoes a non-equilibrium phase transition from a physiological phase with little or no polymer accumulation to a pathological phase characterized by persistent polymerization. A combination of external factors accumulating during the lifetime of a patient can thus slightly modify the phase transition control parameters, tipping the balance from a long symptomless lag phase to an accelerated pathological development.

Ageing phenomena have since a long time been studied, first in several kinds of glassy systems, later on also in more simple systems undergoing cooperative phenomena, such as simple magnets without disorder nor frustrations. The ageing behaviour is seen through an analysis of the two-time correlations and responses. This allows to characterize physical ageing through its three defining properties of (i) slow relaxational dynamics, (ii) breaking of time-transltion-invariance and (iii) dynamical scaling.
Here, the dynamics of surface growth will be analysed through the scaling properties of its two-time observables, where the Edwards-Wilkinson and Kardar-Parisi-Zhang equations will serve as paradigmatic examples.
Specific forms of the fluctuation-dissipation theorem along with consequences for the dynamical scaling will be discussed. The scaling of height and width profiles in semi-infinite systems will be presented in detail.
A new variant of the spherical model of ferromagnets, adapted to surface growth, will be presented and its relation to spherical spin glasses will be derived.

[1] M. Henkel, J.D. Noh, M. Pleimling, Phys. Rev. E 85 030102(R) (2012); [arxiv:1109.5022]
[2] N. Allegra, J.-Y. Fortin, M. Henkel, J. Stat. Mech. P02018 (2014); [arxiv:1309.1634]
[3] M. Henkel, M. Pleimling, "Non-equilibrium phase transitions, vol. 2: Ageing and dynamical scaling far from equilibrium", Springer (Heidelberg 2010)