## Final Report Summary - MOTIONRANDOMMEDIA (Long time behavior for motion in random media)

The project had integrational as well as scientific objectives. We first describe the scientific objectives, and then the integrational ones.

The main scientific objective of the project are to further the understanding of the behavior of random walk in random environment (RWRE) and other models of interacting random walks. More specifically, two main topics have been put forth in the proposal: the first is the understanding of slowdown phenomena in RWRE, and the second is the understanding of slowdown phenomena in other models of interacting random walks.

The work performed since the beginning of the project contains several inter-related main lines of research.

First, we proved that for RWRE in dimension greater than or equal to 4 satisfying Sznitman's ballisticity condition (T') the probability of linear slowdown is bounded above by the negative exponent of the d-epsilon power logarithm of the time, where d is the dimension and epsilon is as small as we want. For every choice of epsilon, this will hold for n large enough.

This upper bound almost matches the known lower bound, which is the negative exponent of the d power logarithm of the time.

and constitutes a significant improvement over the bounds that had been known before. This work was performed by the PI, and the paper reporting it is accepted for publication in the journal of the European mathematical society (JEMS).

We worked on the study of a model of mutually excited random walks (MERW) which, in simulations, showed a surprising phenomenon of non-monotonicity of the limiting velocity as a function of some ``drift''parameter. We proved the existence of a limiting velocity for this process, and constructed an algorithm for rigorously bounding the limiting velocity. While we haven't yet provided a conceptual proof for the non-monotonicity, we have a recipe for a computer-assisted proof for the monotonicity. This work was done jointly by the PI and by the graduate student Eviatar Procaccia. The paper reporting this work is in the final stages of its writing.

We also studied the behavior of random walks on discrete point processes. This was done jointly with my student Ron Rosenthal. A discrete point process is a random subset of Z^d, whose distribution is translation invariant and ergodic. One can then define a random walk on such a process in the following way: Choose, uniformly at random, one of the 2d coordinate directions, and jump to the closest point in that direction. We analyzed this model, and proved an (easy) law of large numbers. We then gave criteria for transience and recurrence. In particular, the walk is always transient in dimension 3 and higher. The most interesting part was a proof of a central limit theorem under the assumption that the distance between neighboring points has a 2+ \epsilon moment. While the main line of this argument follows the by now classic ideas of Barlow and Bass, the unbounded jump distance caused significant technical difficulty, which we solved using an ergodic theorem on free groups, due to Nevo and Stein.

Our paper reporting on these results is currently considered for publication at the Annals Institute Henri Poincare.

Another issue is that of the effect of mixing on convergence to Brownian motion in balanced random environment. This work is based on previous works that proved convergence of random walk on balanced random environments to Brownian motion. The first result, by Lawler, shows that whenever the environment is ergodic and uniformly elliptic, the walk converges to Brownian motion. Two later results, by Guo and Zeitouni and by myself and Deuschel, show that if we require that the environment is i. i. d, we can remove the ellipticity assumption altogether, and replace it by the much weaker assumption of genuine d-dimensionality, namely that there is a positive annealed probability for the first step to be in any direction. Therefore, jointly with my student Moran Cohen, we study the effect of mixing on the non-elliptic environments. In dimension 3 and higher, we have constructed an example of a finite-range dependent system where the CLT fails. In dimension 2 the situation is more complex and depends on the type of mixing. However, for strong enough mixing we can show that the CLT still holds. We also have an example with very weak mixing where the CLT breaks down. The paper with those results is currently being written.

The last issue is the speed of random walk on random conductances. Jointly with Michele Salvi, we studied the following question: Under which circumstances can a random walk on random conductances have a positive speed. We only considered the two-dimensional case. We proved that if the logarithm of the conductance has an expectation, then the speed is always zero. In addition, we found examples where this logarithm had a 1-\epsilon moment, and yet the random walk had positive speed.

The paper reporting this result was submitted for publication at the Annals Institute Henri Poincare.

The integrational objectives were that the researcher integrate in the host institution, and import knowledge from his old old institutions in the USA (i. e. UCLA and Caltech). The integration in the host institution was very successful in the following sense:

(a) 3 years into the project, the researcher was granted tenure, as well as the position of an associate professor in the host institution. This indicates that the host institution viewed the integration as successful.

(b) The researcher had three PhD students (Ron Rosenthal, Eviatar Procaccia, and Moran Cohen) and two Masters students (the same Ron Rosenthal, and Ran Tessler), some of which were supported by the grant.

Transfer of knowledge: The knowledge that the researcher brought with him from the USA was transferred to the host institution in more than one sense:

(a) It was taught to the PhD and Masters students of the researcher.

(b) Parts of it were given as advanced courses for graduate students in the university.

(c) Parts of it were given in research seminars in the university. Most talks were about the researcher's own research, but there were also survey talks on methodology and lines of research which the researcher was exposed to in the USA.

Cooperation with the third country continues. Two examples: My students Ron Rosenthal and Eviatar Procaccia have recently written a joint paper with Marek Biskup, my main collaborator in UCLA. I recently published a paper with Yuval Peres, my PhD advisor at Berkeley.

In particular, potential for leadership and lasting integration have been both demonstrated: Potential of leadership-I built a small group of Masters and PhD students which I successfully lead. I was granted tenure, which shows that I integrated well and will have a lasting relationship with the host institution

The main scientific objective of the project are to further the understanding of the behavior of random walk in random environment (RWRE) and other models of interacting random walks. More specifically, two main topics have been put forth in the proposal: the first is the understanding of slowdown phenomena in RWRE, and the second is the understanding of slowdown phenomena in other models of interacting random walks.

The work performed since the beginning of the project contains several inter-related main lines of research.

First, we proved that for RWRE in dimension greater than or equal to 4 satisfying Sznitman's ballisticity condition (T') the probability of linear slowdown is bounded above by the negative exponent of the d-epsilon power logarithm of the time, where d is the dimension and epsilon is as small as we want. For every choice of epsilon, this will hold for n large enough.

This upper bound almost matches the known lower bound, which is the negative exponent of the d power logarithm of the time.

and constitutes a significant improvement over the bounds that had been known before. This work was performed by the PI, and the paper reporting it is accepted for publication in the journal of the European mathematical society (JEMS).

We worked on the study of a model of mutually excited random walks (MERW) which, in simulations, showed a surprising phenomenon of non-monotonicity of the limiting velocity as a function of some ``drift''parameter. We proved the existence of a limiting velocity for this process, and constructed an algorithm for rigorously bounding the limiting velocity. While we haven't yet provided a conceptual proof for the non-monotonicity, we have a recipe for a computer-assisted proof for the monotonicity. This work was done jointly by the PI and by the graduate student Eviatar Procaccia. The paper reporting this work is in the final stages of its writing.

We also studied the behavior of random walks on discrete point processes. This was done jointly with my student Ron Rosenthal. A discrete point process is a random subset of Z^d, whose distribution is translation invariant and ergodic. One can then define a random walk on such a process in the following way: Choose, uniformly at random, one of the 2d coordinate directions, and jump to the closest point in that direction. We analyzed this model, and proved an (easy) law of large numbers. We then gave criteria for transience and recurrence. In particular, the walk is always transient in dimension 3 and higher. The most interesting part was a proof of a central limit theorem under the assumption that the distance between neighboring points has a 2+ \epsilon moment. While the main line of this argument follows the by now classic ideas of Barlow and Bass, the unbounded jump distance caused significant technical difficulty, which we solved using an ergodic theorem on free groups, due to Nevo and Stein.

Our paper reporting on these results is currently considered for publication at the Annals Institute Henri Poincare.

Another issue is that of the effect of mixing on convergence to Brownian motion in balanced random environment. This work is based on previous works that proved convergence of random walk on balanced random environments to Brownian motion. The first result, by Lawler, shows that whenever the environment is ergodic and uniformly elliptic, the walk converges to Brownian motion. Two later results, by Guo and Zeitouni and by myself and Deuschel, show that if we require that the environment is i. i. d, we can remove the ellipticity assumption altogether, and replace it by the much weaker assumption of genuine d-dimensionality, namely that there is a positive annealed probability for the first step to be in any direction. Therefore, jointly with my student Moran Cohen, we study the effect of mixing on the non-elliptic environments. In dimension 3 and higher, we have constructed an example of a finite-range dependent system where the CLT fails. In dimension 2 the situation is more complex and depends on the type of mixing. However, for strong enough mixing we can show that the CLT still holds. We also have an example with very weak mixing where the CLT breaks down. The paper with those results is currently being written.

The last issue is the speed of random walk on random conductances. Jointly with Michele Salvi, we studied the following question: Under which circumstances can a random walk on random conductances have a positive speed. We only considered the two-dimensional case. We proved that if the logarithm of the conductance has an expectation, then the speed is always zero. In addition, we found examples where this logarithm had a 1-\epsilon moment, and yet the random walk had positive speed.

The paper reporting this result was submitted for publication at the Annals Institute Henri Poincare.

The integrational objectives were that the researcher integrate in the host institution, and import knowledge from his old old institutions in the USA (i. e. UCLA and Caltech). The integration in the host institution was very successful in the following sense:

(a) 3 years into the project, the researcher was granted tenure, as well as the position of an associate professor in the host institution. This indicates that the host institution viewed the integration as successful.

(b) The researcher had three PhD students (Ron Rosenthal, Eviatar Procaccia, and Moran Cohen) and two Masters students (the same Ron Rosenthal, and Ran Tessler), some of which were supported by the grant.

Transfer of knowledge: The knowledge that the researcher brought with him from the USA was transferred to the host institution in more than one sense:

(a) It was taught to the PhD and Masters students of the researcher.

(b) Parts of it were given as advanced courses for graduate students in the university.

(c) Parts of it were given in research seminars in the university. Most talks were about the researcher's own research, but there were also survey talks on methodology and lines of research which the researcher was exposed to in the USA.

Cooperation with the third country continues. Two examples: My students Ron Rosenthal and Eviatar Procaccia have recently written a joint paper with Marek Biskup, my main collaborator in UCLA. I recently published a paper with Yuval Peres, my PhD advisor at Berkeley.

In particular, potential for leadership and lasting integration have been both demonstrated: Potential of leadership-I built a small group of Masters and PhD students which I successfully lead. I was granted tenure, which shows that I integrated well and will have a lasting relationship with the host institution