Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / a / Solution

The Ryu–Takayanagi formula gives the leading entropy of a static boundary region:On the $t=0$ slice of Poincaré ${\mathrm{AdS}}_4$,The minimal surface anchored on a boundary circle of radius $R_1$ is the hemisphereCutting it off at $z=ϵ$, its area isThereforeThe universal requirement was the perimeter-law divergence $\pi R_1/(2Gϵ)$; the finite constant depends on the stated pure ${\mathrm{AdS}}_4$ vacuum geometry and equals $-\pi /(2G)$ here. Since $G\sim N^{-3/2}$, both terms are of order $N^{3/2}$.