March 6, 2015

Location: Achter de Dom 22–24 (Utrecht), room 001

11:15–13:00
Jean-Baptiste Gouéré (MAPMO Orléans) homepage

Continuum percolation on Rd

We consider the Boolean model on Rd. This is the union of i.i.d. random Euclidean balls centered at points of an homogeneous Poisson point process on Rd. Choose the intensity of the Poisson point process so that the Boolean model is critical for percolation. In other words, if we lower the intensity then all the connected components of the Boolean model are bounded, while if we increase the intensity then there exists one unbounded component. We are interested in the volumetric proportion of Rd which is covered by this critical Boolean model. This critical volumetric proportion is a function of the dimension d and of the common distribution of the radii. We aim to study this function.

This is joint work with Régine Marchand.

14:30–16:15
Remco van der Hofstad (TU/e Eindhoven) homepage

Hypercube percolation

Consider bond percolation on the hypercube {0,1}n at the critical probability pc defined such that the expected cluster size equals 2n/3, where 2n/3 acts as the cube root of the number of vertices of the n-dimensional hypercube. Percolation on the hypercube was proposed by Erdős and Spencer (1979), and has proved to be substantially harder than percolation on the complete graph. In this talk, I will describe the phase transition for percolation on the hypercube, and show that it shares many features with that on the complete graph.

In previous work with Borgs, Chayes, Slade and Spencer, and with Heydenreich, we have identified the subcritical and critical regimes of percolation on the hypercube. In particular, we know that for p=pc(1+O(2-n/3)), the largest connected component has size roughly 22n/3 and that this quantity is non-concentrated. In work with Asaf Nachmias, we identify the supercritical behavior of percolation on the hypercube, by showing that, for any sequence εn tending to zero, but εn being much larger than 2-n/3, percolation at pc(1+εn) has, with high probability, a unique giant component of size (2+o(1))εn 2n. This also shows that the proposed critical value really is a correct one.

Finally, we ‘unlace’ the proof by identifying the scaling of component sizes in the supercritical and critical regimes without relying on the lace expansion. The lace expansion is a beautiful technique that is a major technical tool for high-dimensional percolation, but that is also quite involved and can have a disheartening effect on some. For the hypercube percolation phase transition, we no longer need to rely on lace-expansion proofs, nor on results proved elsewhere with the lace expansion. Rather, we use comparisons of percolation paths to non-backtracking random walks.