Extrema of cooling branching Brownian motion and related Gaussian fields

We introduce a family of Gaussian fields with an inhomogeneous cooling variance profile, indexed by $\alpha \in (0,1\mathrm{/}2]$, whose covariance is log–log-correlated at $\alpha =1\mathrm{/}2$ and approaches the log-correlated regime as $\alpha \downarrow 0$. We consider both one dimensional such objects (which we call {\it cooling Branching Brownian Motions}) as well as the related Gaussian fields. We identify the centering of the maximum at the terminal time $T$ and prove tightness of the recentered maximum. While the exponent in the first-order growth varies linearly with $\alpha$, giving a leading order of $T^{1-\alpha }$, the second-order correction exhibits a phase transition at $\alpha =1\mathrm{/}3$. We also show that any subsequential limit cannot be a randomly shifted Gumbel law, in contrast with the logarithmically correlated case.