On a result by Chatterjee on the maximum of a discrete Gaussian
free field.
We consider discrete Gaussian free field (DGFF) on the N-by-N
grid with zero boundary conditions, and with maximum denoted by
X* = X_N*. It is conjectured that Var(X*) = O(1). The current
best result, due to Chatterjee (in "Chaos, concentration, and
multiple valleys") that Var(X*) = o(log N).
More generally, consider a Gaussian process (X_i, i in S),
thought of as one of a family of processes (X^n_i, i in S^n)
with |S_n| tending to infinity. It is easy to show that m =
E[max_i X_i] is bounded above by sqrt[2 s^2 log|S|], where s^2
= max_i Var(X_i); let alpha denote the ratio m/sqrt[2 s^2
log|S|]. In the case of the DGFF, it was shown by Bolthausen,
Deuschel, and Giacomin (AOP 2001) that in fact alpha -> 1. I
will discuss the proof by Chatterjee that for any Gaussian
process for which alpha -> 1, Var(X*) = o(s^2), which in
particular implies his result for the DGFF.