The Page Curve in Higher Dimensions: Phase Transitions, Critical Phenomena, and Finite-Size Broadening

Abstract

Thermodynamic fluctuations smooth Page transitions, but the usual Gaussian description fails at a black-hole critical point. We determine the resulting finite-size crossover in a canonical random-pure-state model whose density of states is specified by the Reissner–Nordström–AdS entropy. In four or more bulk dimensions, the critical fluctuations obey a quartic limiting law, and the transition width grows as the three-quarter power of the critical black-hole entropy. We derive the leading transition profile and its temperature dependence, and identify the reservoir heat capacity required to preserve the canonical critical behavior. Regular-phase examples provide a comparison with Gaussian fluctuations. The entanglement results hold in the planar limit of the specified random-state model; applications to gravitational entropy transitions require identifying the corresponding state and the fluctuating difference of generalized entropies.