Percolation and first-passage percolation – PPPP

We plan to study the mathematical aspects of percolation and first-passage percolation. We do not plan to focus on the critical two-dimensional percolation, which involves specific tools, such as the SLE process introduced by Schramm in 1999, and was for example the topic of ANR MAC2 (ANR-10-BLAN-0123).

Suppose we immerse a large porous stone in a bucket of water. Consider the following closely related problems:
- What is the probability that the center of the stone is wetted ?
- What is the time needed for the center of the stone to be wetted ?

Percolation is a model for the first problem. This is the study of connectedness properties of random graphs. Edges model pairs of close points of the stone between which the water can flow. It was introduced by Broadbent and Hammersley in 1957.

First-passage percolation is a model for the second problem. This can be seen as the study of random metrics. The random distance between two points models the time needed for the water to flow from one point to the other. It can also be seen as a model of random growth or as a model of competition. It was introduced by Hammersley and Welsh in 1965.

The models are closely related by their mere definition (first passage percolation is a refinement of percolation) but also because their study share many ideas and tools (coarse graining, coupling, FKG and BK inequalities, ...). The theory of these models is well developed, at least in the standard case (i.i.d. case on Z^d). However, there remains several intriguing and important questions and long-standing conjectures. Recent significant developments on these issues, some of which involving members of the team, give some real hope to make further significant progress on these problems.

We see the conjectures as stimulations for our research. We aim to make some progress on these problems but, more generally, our aim is to investigate first-passage percolation, percolation and their links both in the standard case and in some less standard ones.