An M-dependent process X(n) on the integers, is a process for which every event concerning with X(-1),X(-2),... is independent from every event concerning with X(M),X(M+1),...

Such processes play an important role both as scaling limits of physical systems and as a tool in approximating other processes.

A question that has risen independently in several contexts is:

"is there an M dependent proper colouring of the integer lattice for some finite M?"

Ramsey theory guarantees that the most standard way of producing an M-dependent process can never produce such a colouring. That is, defining X(i)=f(Y(i),Y(i+1),...,Y(i+M)) where Y(n) are uniform([0,1]) i.i.d. (such processes are called M+1-block factors). It was therefore conjectured to be impossible.

After a short introduction to the history of the problem, I will describe a recent simple construction of Holryd and Liggett refuting this conjecture. Giving a 1-dependent proper 4-colouring and a 2-dependent proper 3-colouring of the integer lattice, which are produced in an entirely new manner via an insertion process.

## Date:

Tue, 02/01/2018 - 12:00 to 13:00