We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate $\mu$, while at the same time a random walker moves on $G$ at rate 1, but only along edges which are open. On the d-dimensional torus with side length $n$, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, Stauffer and Steif. I will talk about the supercritical case, which was left open, but can be analysed using evolving sets.

Joint work with Y. Peres and J. Steif.[-]

In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of different step distributions needed to obtain a recurrent process. In the second, unrelated, half of the talk, I will present joint work with Tom Hutchcroft, showing that the component structure of the uniform spanning forest in $\mathbb{Z}^d$ changes every dimension for $d > 8$. This sharpens an earlier result of Benjamini, Kesten, Schramm and the speaker (Annals Math 2004), where we established a phase transition every four dimensions. The proofs are based on a the connection to loop-erased random walks.[-]

Motivated by solgel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing ('freeze') as soon as they become infinite.
In this talk I will discuss recent (and ongoing) work, with Demeter Kiss and Pierre Nolin, on processes of similar flavour on planar lattices. We focus on the problem whether or not the giant (i.e. 'frozen') clusters occupy a negligible fraction of space. Accurate results for near-critical percolation play an important role in the solution of this problem.
I will also present a version of the model which can be interpreted as a sensor/communication network.[-]