HOẠT ĐỘNG TRONG TUẦN

Percolation on graphs with polynomial growth
Speaker: Sébastien Martineau, Sorbonne Université (Jussieu, Paris 6)

Thời gian: 14h Thứ 5, ngày 29/09/2022

Địa điểm: Phòng 507 nhà A6

Link online Zoom: 845 8621 8812

Passcode: 692956

Tóm tắt: Take a graph, such as the square lattice in dimension 2 or the cubic lattice in dimension 3, and fix some parameter p in [0,1]. Now, independently for each edge, erase it with probability 1-p and keep it with probability p. How does this random graph typically look like? Percolation theory is devoted to answering this question.

More precisely, we are interested in the connected components of this graph, which are called clusters in percolation jargon. Are there infinite clusters? How many? If we focus on finite clusters, are they typically very small or can they get pretty large?

I will devote time to introducing the model and explaining what researchers in the field are interested in (theorems, conjectures). Then, I will present some works made in collaboration with Daniel Contreras and Vincent Tassion.

In these works, we extend the understanding of percolation from d-dimensional cubic lattices to a more general class of graphs. Instead of working with explicit graphs, we only need two assumptions:
1. (homogeneity) the graph looks the same seen from any of its vertices;
2. (polynomial growth) the cardinality of the ball of radius n is upper-bounded by a polynomial in n.

This extension is the opportunity to develop more robust arguments. If we want to stay with d-dimensional cubic lattices, we do not prove new theorems but we provide new proofs of known results.

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