MARK KAC SEMINAR

We consider the Boolean model on R^d. This is the union of i.i.d. random Euclidean balls centered at points of an homogeneous Poisson point process on R^d. 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 R^d 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.

Consider bond percolation on the hypercube {0,1}ⁿ at the critical probability p_c defined such that the expected cluster size equals 2^n/3, where 2^n/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=p_c(1+O(2^-n/3)), the largest connected component has size roughly 2^2n/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 ε_n tending to zero, but ε_n being much larger than 2^-n/3, percolation at p_c(1+ε_n) has, with high probability, a unique giant component of size (2+o(1))ε_n 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.