Extended Data Fig. 1: Action difference - Large black hole.
Results for four spatial dimensions (d = 4). We set the outer horizon radius to be r+ = 100, with the AdS length given by LAdS = 1. a, We chose r− = 50 for the inner horizon radius. When the tension T is not too close to its critical value Tcrit = 1/LAdS = 1, the preparation time τ0 is positive (i.e. the solution is Euclidean-sensible) and the action difference ΔI is negative, meaning that the non-extremal phase is always dominant in the path integral. b, For a near-critical brane (we chose a value T = 0.99999 ~ Tcrit for the brane tension), when the black hole is sufficiently close to extremality (i.e. r− → r+), the preparation time τ0 becomes positive, and therefore the Euclidean solution is sensible. The non-extremal phase is already dominant in the ensemble (i.e. the action difference ΔI is negative) well before it happens. The same properties hold also for the small black hole case.
