## Periodic Reporting for period 1 - EinsteinVRH (Einstein Relation for the Variable Range Hopping model)

**Reporting period:**2016-02-01

**to**2018-01-31

## Summary of the context and overall objectives of the project

SUMMARY

Semiconductors form the basis for modern electronic and a study of their electrical conductivity is paramount both for applications and for our comprehension of their structure. A model for the flux of electrons in this kind of materials, called the Variable-Range Hopping, has been proposed in the 70’s from the physics community and it is still used nowadays. In mathematical terms, it accounts to a long-range reversible stochastic processes on a random point process, the point process representing the impurities of the material where the electrons localize. For experimental purposes, physicists often have to assume the validity of the Einstein relation for this model, i.e. the equivalence between the diffusivity of the electrons and their mobility.

Rigorously proving the Einstein relation for a stochastic process requires an extremely accurate control of the its asymptotics, and for this reasons in the mathematical literature authors had always to assume rather restrictive theoretical hypothesis. Our main goal is to give a completely rigorous proof of the first Einstein relation for a realistic model: the Variable-Range Hopping.

Besides the importance for its applications in physics, the project aims at attacking the tough and relatively unexplored framework of random walks in a random environment with infinite-range. The classical theory does not cover this kind of processes, and new tools have to be invented to face them. These tools give in turn the possibility of approaching other models from a new perspective.

CONCLUSION OF THE ACTION

The project “EinsteinVRH” has been successful beyond expectations. Besides accomplishing the main goal (the proof of the Einstein relation for the Variable-Range Hopping model), our work lead to new techniques and tools that we applied to other models. In the two years of the action, five between publications and submitted articles have been produced and several new projects have started. Our results have been disseminated through conferences, seminars and discussions with other academics, raising the interest of the experts and attracting other scientists on the topic.

Semiconductors form the basis for modern electronic and a study of their electrical conductivity is paramount both for applications and for our comprehension of their structure. A model for the flux of electrons in this kind of materials, called the Variable-Range Hopping, has been proposed in the 70’s from the physics community and it is still used nowadays. In mathematical terms, it accounts to a long-range reversible stochastic processes on a random point process, the point process representing the impurities of the material where the electrons localize. For experimental purposes, physicists often have to assume the validity of the Einstein relation for this model, i.e. the equivalence between the diffusivity of the electrons and their mobility.

Rigorously proving the Einstein relation for a stochastic process requires an extremely accurate control of the its asymptotics, and for this reasons in the mathematical literature authors had always to assume rather restrictive theoretical hypothesis. Our main goal is to give a completely rigorous proof of the first Einstein relation for a realistic model: the Variable-Range Hopping.

Besides the importance for its applications in physics, the project aims at attacking the tough and relatively unexplored framework of random walks in a random environment with infinite-range. The classical theory does not cover this kind of processes, and new tools have to be invented to face them. These tools give in turn the possibility of approaching other models from a new perspective.

CONCLUSION OF THE ACTION

The project “EinsteinVRH” has been successful beyond expectations. Besides accomplishing the main goal (the proof of the Einstein relation for the Variable-Range Hopping model), our work lead to new techniques and tools that we applied to other models. In the two years of the action, five between publications and submitted articles have been produced and several new projects have started. Our results have been disseminated through conferences, seminars and discussions with other academics, raising the interest of the experts and attracting other scientists on the topic.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

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.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project has been extremely successful in both achieving the main goals that were expected and in opening new stimulating paths of research.

I am continuing to work on the subjects of the “EinsteinVRH” project together with Alessandra Faggionato and Nina Gantert. We are analyzing the question of the Einstein relation for the Mott random walk in higher dimensions and for other processes in random environment. We are also particularly interested in the so called Einstein relation out of equilibrium, where one studies the mobility and the diffusivity of the system for values of the external field different from the critical one. In April the three of us will be hosted at the CIRM institute in Marseille for a workshop where a number of experts in the field of random processes in random media and in particular of the Einstein relation will discuss the state of the art in the mathematical community.

Together with Scott Armstrong (NYU, New York, US) I am currently working on a counterexample showing that -contrary to the common belief in the physical and mathematical communities- the Einstein relation is not in general valid for an elliptic and shift invariant environment. We borrow techniques from quantitative homogenization combined with the ideas present in an old work of mine in collaboration with Noam Berger (Jerusalem University).

With Quentin Berger we are still working on sub-ballistic random walks among biased random conductances. We aim at proving the convergence of the whole process to the inverse of an alpha-stable subordinator, with the same alpha found in our previous paper. From there we would like to go a step further and generalize the notion of Einstein relation to include also sub-ballistic processes, which has never been done.

I am continuing to work on the subjects of the “EinsteinVRH” project together with Alessandra Faggionato and Nina Gantert. We are analyzing the question of the Einstein relation for the Mott random walk in higher dimensions and for other processes in random environment. We are also particularly interested in the so called Einstein relation out of equilibrium, where one studies the mobility and the diffusivity of the system for values of the external field different from the critical one. In April the three of us will be hosted at the CIRM institute in Marseille for a workshop where a number of experts in the field of random processes in random media and in particular of the Einstein relation will discuss the state of the art in the mathematical community.

Together with Scott Armstrong (NYU, New York, US) I am currently working on a counterexample showing that -contrary to the common belief in the physical and mathematical communities- the Einstein relation is not in general valid for an elliptic and shift invariant environment. We borrow techniques from quantitative homogenization combined with the ideas present in an old work of mine in collaboration with Noam Berger (Jerusalem University).

With Quentin Berger we are still working on sub-ballistic random walks among biased random conductances. We aim at proving the convergence of the whole process to the inverse of an alpha-stable subordinator, with the same alpha found in our previous paper. From there we would like to go a step further and generalize the notion of Einstein relation to include also sub-ballistic processes, which has never been done.