Our analysis aims at providing rigorous mathematical results to several models in Euclidean quantum fields theory and classical statistical mechanics with relevant physical interest. We analyzed systems defined both in the continuum and on the lattice.
     The aspects we have mostly considered are related to the following questions: renormalization and the continuum limit, stability bounds and the thermodynamical limit, the energy-momentum spectrum and the transfer matrix spectrum, decay properties of correlations. The mathematical tools we use include analysis, functional analysis, spectral analysis, analytical expansions (polymer, cluster and Mayer expansions) and renormalization group maps and techniques.

      One of the most interesting challenges in Network Science today is to understand the relation between the structure of the system and its emergent dynamical properties. For instance, if we know how the structure of networks influences epidemic spreading, then we can predict the diffusion of diseases on a society and develop methods to control the pathogen propagation. The network structure is also fundamental to understand the synchronization phenomenon, which is a ubiquitous process in natural and artificial systems. This dynamical process can model the function of neurons in the central nervous system, power stations, crickets, heart cells and lasers. Thus, we can understand and control the behavior of several natural and artificial systems if we know how the network structure impacts the synchronization of coupled oscillators. Numerous other dynamical processes, such as cooperation and cascade failures, can also be studied in networks.
      Our research today focuses on the study of the influence of network properties on stochastic and dynamical processes. We are interested in answering the following questions: What are the universal patterns of connections in networks or in classes of networks? How does the network organization influence the spreading of diseases or information propagation? How is the effect of local and large-scale properties of networks on the synchronization? How can we infer the network properties from sampled networks? In addition, we are interested in several applications, including financial market, pattern recognition and data mining, neuroscience, ecology and power grids.

      In one dimensional systems, long range models are a major source of problems. "Long range" means "not necessarily Markovian". Our interest is in large part due to the fact that they account for many interesting statistical physics phenomenon, such as phase transition (which cannot occur in finite range models), Gibbs/non-Gibbs transitions etc. We are specially interested in some basic questions such as existence, uniqueness, and statistical property (perfect simulation algorithms, mixing rate, limit theorem etc) of the stationary measure. But also in statistical inference of some properties of these chains, and applications to real datas (linguistic and neuroscience for example).
      Another topic of interest is the study of hitting and return time distributions. In stochastic processes and dynamical systems, there is the Poincaré Reccurrence Theorem, a very classical result establishing that almost every realization of the process hits any fixed observable, infinitely many times. Since then, many works consider the problem of presenting quantitative information for these occurrences and other related quantities regarding the statistical analysis of observables. In particular, the right rescalling to the exponential law for hitting/return times is a very extensively studied topic in the literature nowadays.

      We apply interacting particle systems, percolation models, and special stochastic processes to define theoretical models describing the spread of an information, or similar phenomena, on a population. The considered stochastic models are variants of the classical SIS (susceptible-infected-susceptible), known to probabilists as the contact process, and SIR (susceptible-infected-removed) epidemic models. We investigate the asymptotic behavior, and the existence of phase transitions, of such processes on different (random) graphs.