In this thesis, exit-time problem for two types of non-linear diffusion processes is considered. The first process is called the Self-interacting diffusion and is defined by the following SDE including interaction of the process with its own path: d X_t = \sigma d W_t - ( \nabla V(X_t) + \frac{1}{t} \int_0^t \nabla F (X_t - X_s) d s) d t.The second...
Published in New Journal of Physics
We numerically investigate the dynamics of an SIR model with infection level-based lockdowns on Small-World networks. Using a large-deviation approach, namely the Wang–Landau algorithm, we study the distribution of the cumulative fraction of infected individuals. We are able to resolve the density of states for values as low as 10−85. Hence, we mea...
We study the wetting model, which considers a random walk constrained to remain above a hard wall, but with additional pinning potential for each contact with the wall. This model is known to exhibit a wetting phase transition, from a localized phase (with trajectories pinned to the wall) to a delocalized phase (with unpinned trajectories). As a pr...
Published in Journal of Statistical Mechanics: Theory and Experiment
The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force is characterized by explicit dynamics of their integer moments and by explicit relaxation of spectral properties towards their steady state. Besides the Ornstein–Uhlenbeck process with a Gaussian steady state, other representative examples of the Pear...
Published in Journal of Statistical Mechanics: Theory and Experiment
We present the application of a fluctuating hydrodynamic theory to study current fluctuations in diffusive systems on a semi-infinite line in contact with a reservoir with slow coupling. We show that the distribution of the time-integrated current across the boundary at large times follows a large deviation principle with a rate function that depen...
Published in Journal of Statistical Mechanics: Theory and Experiment
Stochastic processes with random reinforced relocations have been introduced in a series of papers by Boyer and co-authors (Boyer and Solis Salas 2014, Boyer and Pineda 2016, Boyer, Evans and Majumdar 2017) to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at ra...
Published in Journal of Physics A: Mathematical and Theoretical
We simulate spreads of diseases for the susceptible–infected–recovered (SIR) model on contact networks with a particular focus on incorporated protective measures such as face masks. We consider the small-world network model. By using a large-deviation approach, in particular the 1/t Wang–Landau algorithm, we obtained the full probability density f...
Published in Journal of Statistical Mechanics: Theory and Experiment
For boundary-driven non-equilibrium Markov models of non-interacting particles in one dimension, either in continuous space with the Fokker–Planck dynamics involving an arbitrary force F(x) and an arbitrary diffusion coefficient D(x), or in discrete space with the Markov jump dynamics involving arbitrary nearest-neighbor transition rates w(x±1,x) ,...
Published in Journal of Physics A: Mathematical and Theoretical
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the...
Published in Discrete Mathematics and Applications
We consider the branching process Zn=Xn,1+⋯+XnZn−1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $, in random environmentsη, where η is a sequence of independent identically distributedvariables, for fixed η the random variables Xi, j areindependent, have the geometric distribution. We suppose that the associated random walk Sn=ξ1+⋯+ξn $ S_n = \xi_1 + \...
