TCC Course: (Atypical) behaviour of random walks in random or dynamic environment.

I will be posting here the lecture notes for my TCC course on trapping phenomena for random walks in random environment, for graduate students at Bath, Bristol, Imperial College London, Oxford, Swansea and Warwick.

The course is on MS Teams, every Thursday from 2pm to 4pm, for 8 weeks starting from the week of October 14th.

Synopsis: We will be interested in trapping phenomena that can occur in certain random walks in random environment. These are often due to areas in the environment which behave in an atypical manner: even though these areas can be small, i.e. microscopic, their impact is important as it can change the macroscopic behaviour of the random walk, for instance by dramatically reducing its speed or by damping its diffusive behaviour into sub-diffusivity. These phenomena are closely related to a model from physics called the Bouchaud Trap Model. 

We will go over several results and questions around these topics, including the one-dimensional and multi-dimensional random walk in random environment, as well as random walks on Galton-Watson trees. We may investigate related questions about the monotonicity of the speed in some of these models. Finally, if time allows, we will discuss similar questions about random walks in dynamical random environment.

Lecture 1: Find my notes here. References:

-« Topics in random walk in random environment », Alain-Sol Sznitman, find it here;

-« Ten lectures on random media», E. Bolthausen and A.S. Sznitman, find it here.

Wikipedia is a good source for some of the basics I recalled, like the Law of Large Numbers, Central Limit Theorem, Donsker’s invariance principle, Birkhoff’s Ergodic Theorem and the Pointwise Ergodic Theorem.

Lecture 2: Notes are here. References:

-« Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension » by Fontes, Isopi and Newman, proving the convergence of the Bouchaud trap model towards the FIN diffusion, here;

-« Aging and sub-aging for one-dimensional random walks amongst random conductances » by Croydon, Scali and myself, where we prove diverse results on the random walk on random conductances I discussed, here;

-« Scaling limits of stochastic processes associated with resistance forms », lecture notes by Croydon where he discussed scaling limits and resistance space, here.

Lecture 3: Notes are here. References:

-« Aging and sub-aging for one-dimensional random walks amongst random conductances » by Croydon, Scali and myself, where we prove diverse results on the random walk on random conductances I discussed, here;

-« Scaling limits of stochastic processes associated with resistance forms », lecture notes by Croydon where he discussed scaling limits and resistance space, here.

-« Randomly trapped random walks », by Ben Arous, Cabezas, Cerny and Royfman, for the random walks on the comb graphs, here;

-« Topics in random walk in random environment », Alain-Sol Sznitman, find it here;

-« Ten lectures on random media», E. Bolthausen and A.S. Sznitman, find it here.

Lecture 4: Notes are here. References:

-« Random walks in random environment», F. Solomon, find it here.

-« The limiting behavior of a one-dimensional random walk in a random medium», Ya. G. Sinai, find it here.

-« The limit distribution of Sinai’s random walk in random environment», H. Kesten, find it here.

Lecture 5: Notes are here. References:

-« Topics in random walk in random environment », Alain-Sol Sznitman, find it here;

-« On a class of transient random walks in random environment », Alain-Sol Sznitman, find it here;

-« Local trapping for elliptic random walks in random environments in Z^d », A. Fribergh and D. Kious, find it here;

Lecture 6: Notes are here. References:

-« Random walk and percolation on trees », R. Lyons, find it here;

-« Biased random walks on Galton-Watson trees », Lyons, Pemantle and Peres, find it here.

Lecture 7: Notes are here. References:

-« Random walk and percolation on trees », R. Lyons, find it here;

-« Biased random walks on Galton-Watson trees », Lyons, Pemantle and Peres, find it here.

-« Random walk on the simple symmetric exclusion process », Hilàrio, Kious and Teixeira, here.

-« Sharp threshold for the ballisticity of the random walk on the exclusion process », Conchon—Kerjan, Kious and Rodriguez, here.

Lecture 8: Notes are here. References:

-« Random walk on the simple symmetric exclusion process », Hilàrio, Kious and Teixeira, here.

-« Sharp threshold for the ballisticity of the random walk on the exclusion process », Conchon—Kerjan, Kious and Rodriguez, here.