Probability distribution of time of detection of a quantum-mechanical particle at a screen

Imagine an experiment where a quantum-mechanical particle is released from
some fixed region inside a box. On one side of the box there is a screen
with detectors which click as soon as the particle "arrives" at the screen. One expects that the time of arrival of the particle is a stochastic variable and it is interesting to ask for it's probability distribution.
This is similar to asking for the distribution of the time of  absorption of Brownian particle at some point. In this talk, an attempt will be made to explain why the quantum  problem is subtle, and our recent attempts at understanding this in a framework where  repeated projective  measurements
are made to detect the particle. This leads to  a non-unitary time evolution
of the wave-function of the particle, and we show that this is well described by an effective non-Hermitiian  Hamiltonian. For  some simple lattice models, we find  power-law tails for the probability that the particle  survives detection up to some time.