Explicit mechanism for Hawking–Page tunneling

Develop an explicit semiclassical or quantum-gravitational description of the tunneling process by which a thermal gas in asymptotically Anti de Sitter space transitions to an AdS–Schwarzschild black hole (and vice versa) across the Hawking–Page transition, including the relevant instanton or real-time mechanism and its rate.

Background

In asymptotically AdS spacetimes, at high temperatures the dominant phase is an AdS–Schwarzschild black hole, while at lower temperatures thermal AdS dominates. Near the transition, one expects tunneling between these phases, suggesting a nonperturbative process akin to bubble nucleation or instanton-mediated transitions.

While the thermodynamic dominance and phase structure are well understood, a concrete description of the tunneling process itself—either as a Euclidean instanton or a Lorentzian real-time mechanism—has not been established in this context.