Logarithmic variance for uniform homomorphisms on Z^2
Take the n by n box in the square lattice and fill it with integers where the integers on adjacent vertices only differ by +1 or -1. How large can a typical value be?
We prove that the variance at a point blows up like log n which in particular implies that such a function is delocalized. This is a consequence of a Russo Seymour Welsh theory that we build for such functions. I will also discuss a connection with the six-vertex model (also called the square ice model).
Joint work with Hugo Duminil Copin, Matan Harel, Benoit Laslier and Aran Raoufi.