An Analytic Model with Critical Behavior in Black Hole Formation
  A simple analytic model is presented which exhibits a critical behavior in black hole formation, namely, collapse of a thin shell coupled with outgoing null fluid. It is seen that the critical behavior is caused by the gravitational nonlinearity near the event horizon. We calculate the value of the critical exponent analytically and find that it is very dependent on the coupling constants of the system. 
Tatsuhiko Koike and Takashi Mishima