Multifractal random walks (MRW) correspond to simple solvable “stochastic volatility” processes. Moreover, they provide a simple interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that they are able to...

We establish a “Central Limit Theorem” for rank distributions, which provides a detailed characterization and classification of their universal macroscopic statistics and phase transitions. The limit theorem is based on the statistical notion of Lorenz curves, and is termed the “Lorenzian...

The mode-coupling equations used to study glasses and supercooled liquids define the underlying regenerative processes represented by an indicator function Z(t). Such a process is a special case of an alternating renewal process, and it introduces in a natural way a stochastic two level system....

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For...

In the realm of multiscale signal analysis, multifractal analysis provides a natural and rich framework to measure the roughness of a time series. As such, it has drawn special attention of both mathematicians and practitioners, and led them to characterize relevant physiological factors...

The Onsager linear relations between macroscopic flows and thermodynamics forces are derived from the point of view of large deviation theory. For a given set of macroscopic variables, we consider the short-time evolution of near-equilibrium fluctuations, represented as the limit of finite-size...

We present a complete analytical solution of a system of Potts spins on a random k-regular graph in both the canonical and microcanonical ensembles, using the Large Deviation Cavity Method (LDCM). The solution is shown to be composed of three different branches, resulting in a non-concave...

The criterion of minimizing the cumulative hedged returns’ probability of underperforming a benchmark provides a framework for evaluating short-term hedges that are rolled over to produce longer-term hedges. Large deviations theory can be used to either parametrically or nonparametrically...

A recent theory by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim predicts a temporal asymmetry in the fluctuation–relaxation paths of certain observables of nonequilibrium systems in local thermodynamic equilibrium. We find temporal asymmetries in the fluctuation–relaxation paths of a...

In these lectures, we shall present some remarkable results that have been obtained for systems far from equilibrium during the last two decades. We shall put a special emphasis on the concept of large deviation functions that provide us with a unified description of many physical situations....