Conformal Invariant Planar Processes

This project deals with a generalization of percolation interfaces to higher dimensions. Critical percolation on the faces of the hexagonal lattice has been thoroughly studied and there have been great advances using to Smirnovls parafe7mi()nic observables and Schramm’s SLE process. These constructions are very specific to two dimensions.

Higher dimensional behavior is much more mysterious. We define a multi-color percolation model generalizing the interfaces of percolation on the hexagonal lattice. The underlying structure is the perrnutahrzdral lattice, a tessellation of R4 with cells a a particular feature: if we consider the vertices and edges of these cells, every vertex lies on the intersection of d + 1 pairwise incident cells, and every edge is the intersection of d pairwise incident cells, where the endpoints of the edge correspond to adding one of two possible cells to form d + 1 pairwise incident cells.

If we color the cells in d colors, this structure enables us to define multi-tzolored edges: those edges that lie on the intersection of cells colored in d different colors. Because of the lattice structure, the collection of such edges forms a collection of simple cycles and possibly bi-infinite simple paths. We then color every cell with one of d colors independently, each color j with probability pj, where p : (p1,. . . ,pd) is a probability vector. The main question of interest is the size of typical multi-colored cycles and paths in the resulting coloring.

We were able to prove that the probability simplex {II E R4 : pl 2 U , 2] pl : 1} is divided into two phases: the extended phase and the compact phase. In the compact phase the probability of a long multi-colored path decays exponentially in the length. This phase corresponds to the sub-critical phase in percolation. In the extended phase, there are long multi-colored paths at all scales.

Using a novel argument we were able to show that the extended phase is not empty.

Studying the properties of this model in three dimensions we were also able to understand other interfaces between the three colors. Of course, in order for there to be infinite (or “long”) tricolored paths, it is necessary that each one of the three colors is not sub-critical. One may naively conjecture that as long as all three colors are super-critical there exist infinite tricolored paths (because the infinite components of all three colors must somehow create a mutual interface). However, we show that this is not the case, and there is a phase of three co-existing infinite components, one of each color, but no infinite (or even “long”) tricolored path.

In addition to gaining knowledge of the model itself, we were also able to apply this to obtain a non-trivial bound on the critical parameter for site percolation on the three-dimensional permutahedral lattice.