I will give a brief overview of very simple, hence maybe less investigated structures in interacting particle systems: reversible product blocking measures. These turn out to be more general than most people would think, in particular asymmetric simple exclusion and nearest-neighbour asymmetric zero range processes both enjoy them. But a careful look reveals that these two are really the same process. Exploitation of this fact gives rise to the Jacobi triple product formula - an identity previously known from number theory and combinatorics. I will show you the main steps of deriving it from pure probability this time, and I hope to surprise my audience as much as we got surprised when this identity first popped up in our notebooks.

We prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension d>4 undergoes a non-trivial phase transition (in the sense that p_c<1). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique consisting in expressing certain functionals of the Gaussian Free Field (GFF) in terms of connectivity probabilities for percolation model in a random environment. Then, we integrate out the randomness in the edge-parameters using a multi-scale decomposition of the GFF. We believe that a similar strategy could lead to proofs of the existence of a phase transition for various other models.
Joint work with Hugo Duminil-Copin, Subhajit Goswami, Franco Severo, Ariel Yadin.

It is classical that the zero set and the set of record times of a linear Brownian motion have Hausdorff dimension 1/2 almost surely. Can we find a larger random set on which a Brownian motion is monotone? Perhaps surprisingly, the answer is negative. We outline the short proof, which is an application of Kaufman's dimension doubling theorem for planar Brownian motion. If time permits, we discuss related results for random walk and fractional Brownian motion as well, and pose some open problems. This is a joint work with Omer Angel, András Máthé, and Yuval Peres.

The topic of this talk will be random permutations coming from the interchange process and its generalizaton, the quantum Heisenberg model from statistical physics, on the 2-dimensional Hamming graph. We prove the existence of a phase transition - in the interchange process for sufficiently long times there exists, with high probability, a cycle of macroscopic size, while for short times all cycles are small. This is the first result of this kind for a graph with nontrivial geometry, generalizing known results due to Schramm (2008). One of the main techniques used is the cyclic time random walk. Joint work with Radosław Adamczak and Piotr Miłoś.