  Consider a particle sitting at a fixed position outside of a stable black
hole. If the system is heated up, the black hole horizon grows and there should
exist a critical temperature above which the particle enters the black hole
interior. We solve a simple model describing exactly this situation: a large N
matrix quantum mechanics modeling a fixed D-particle in a black hole
background. We show that indeed a striking phenomenon occurs: above some
critical temperature, there is a non-perturbative Bose-Einstein condensation of
massless strings. The transition, even though precisely defined by the presence
of the condensate, cannot be sharply detected by measurements made in a finite
amount of time. The order parameter is fundamentally non-local in time and
corresponds to infinite-time correlations.
