Numerical schemes for dynamical mean-field theory on finitely-connected graphs.

The cavity method is an effective technique born to analyze classical system defined on graphs, and then also quantum systems, at equilibrium; by exploiting graphical models, it is indeed possible to derive self-consistent equations for one-site cavity marginals that, if obtained, allow for the computation of full one-site marginals, so that local observables can be easily computed. At equilibrium, and especially in the case of quantum systems, the results obtained with the cavity method provide a starting point for the application of the dynamical mean-field theory approach: this allows for a description of a many-body problem as a one-body problem by means of a set of effective quantities that have to be computed self-consistently. The dynamical mean-field theory approach has proven its effectiveness in many-body quantum mechanics for the analysis of both fermionic (F-DMFT) and bosonic (B-DMFT) systems. The approach based on the cavity method and on dynamical mean-field theories for the analysis of classical and quantum systems at equilibrium can be extended to out-of-equilibrium classical systems; these can be defined as a set of stochastic differential equations on a graph, each of them describing the behavior of a degree of freedom associated to a node subjected to a local term, to the interactions with the neighbors and to an additive noise. Again with the help of graphical models, it is possible to apply the dynamic cavity method to derive a set of self-consistent equations for the cavity marginals and then for the full marginals; by performing a large connectivity expansion typical of the dynamical mean-field theory approach, one is able to obtain a set of effective stochastic differential equations, one for each degree of freedom, where the interaction term gets substituted by a set of terms involving cavity mean functions, cavity correlation functions and cavity response functions. The set of effective equations can be reduced to a single effective equation, with a single cavity mean function, a single cavity correlation function and a single response function, if one considers regular graphs, like a Bethe lattice; it is also possible to analyze disordered systems with this approach by performing configurational averages. The goal of this thesis is the development of an algorithm aimed at computing the cavity mean function, the cavity correlation function and the cavity response function appearing in the effective stochastic differential equation of a generic dynamics defined on a Bethe lattice with linear interactions and additive noise. From a conceptual point of view the structure of the algorithm is the following: the cavity quantities are initialized, then they are used to generate a certain number of trajectories according to the effective equation of the dynamics, which in turn are used to update the cavity quantities in an iterative fashion until convergence is reached. After convergence, the cavity quantities can be plugged into the effective equation of the dynamics, so that trajectories can be generated in order to perform a statistical analysis of the dynamics under exam.