We describe here the obtained results.
1. The velocity of 1D Variable-Range Hopping with external field, in collaboration with Alessandra Faggionato (La Sapienza university, Rome) and Nina Gantert (TUM, Munich), accepted in Annales de l’Institut Poincaré.
In this paper, we have begun the study of the asymptotic speed of the Variable-Range Hopping model, or Mott random walk, under the influence of an external field in dimension d = 1. We have given sharp explicit conditions on the disordered environment in order to have zero or non-zero limiting velocity.
I have presented these results in several conferences and seminars, as the Probability World Congress in Toronto, the MIPS seminar at Leiden University (Netherland).
2. Einstein relation and linear response in one-dimensional Matt Variable-Range Hopping, in collaboration with Alessandra Faggionato (La Sapienza university, Rome) and Nina Gantert (TUM, Munich)
Thanks to a functional-analytic technique suggested by the supervisor of the fellowship Stefano Olla and thanks to the results achieved in the first step, we have proven the Einstein relation for the Variable-Range Hopping model in dimension 1. The idea is to study the integrability of the steady states of the process, finding bounds that are uniform in the intensity of the external field, and to apply the theory developed by Donsker and Varadhan in the ‘80s. This new technique has the potential to be applied to other models, too.
I have presented these results in several conferences and seminars, as the 39th Conference on Stochastic Processes and their Applications in Moscow (Russia), the probability seminars of the UPEC university and of the École Polytechnique in Paris.
3. Scaling of sub-ballistic 1d random walks among biased random conductances, in collaboration with Quentin Berger (Université Jussieu, Paris)
We analyze random walks perturbed by an external field, in the case where its intensity is not sufficiently strong for having a strictly (say) positive limiting velocity, but the walk is still transient. We find the right rescaling of the walk for two different models, the classical random conductance model and the range-one Mott walk. Interestingly, the rescaling exponent for the first does not depend on the intensity of the external field, whereas the second does.
4. Regularity of biased 1d random walks in random environment, in collaboration with Alessandra Faggionato (La Sapienza, Roma)
We consider general random walks in random environment with an external field. We study the regularity properties (continuity, monotonicity, analiticity…) of the asymptotic velocity and of the diffusivity of the process as functions of the intensity of the field. We also extend the known results about the Einstein relation for the random conductance model.
5. Random walk on a perturbation of the infinitely-fast mixing interchange process, in collaboration with François Simenhaus (Paris-Dauphine), accepted in Journal of Statistical Physics.
Random walks in dynamic random environment have gained much attention from the mathematical community in the recent years. If the environment is given by the interchange process (a generalization of the exclusion process) the traps present in the medium might remain for a long time. We prove instead that if the underlying dynamics mixes fast enough, the empirical velocity of the process approaches the annealed one.
The result has been presented in several conferences and seminars by F. Simenhaus.
