Extrema of microscopically slowed-down Gaussian fields

Abstract

We introduce a family of Gaussian fields whose covariance structure exhibits an inhomogeneous, microscopic slowdown and it interpolates between a $\log$ profile (for a certain interpolation parameter $α=0$) and a $\log\log$ profile (when the interpolation parameter is $α=1/2$). We consider both one dimensional such objects (which we call {\it Branching Brownian Motions in a cooling environment}) as well as higher dimensional, spatial fields. We identify the correct centering of the maximum at time $T$ and prove tightness of the recentered maximum. While the exponent in the first-order growth varies linearly with $α$, giving a leading order of $T^{1-α}$, the second-order correction exhibits a phase transition at $α=1/3$.