Oh I see, I get it now. Now I just don't understand why we use only one pair of of fields to estimate this correction factor. Luescher's notes (at least the RHMC ones) seem to indicate that one should really take a stochastic estimate. (I realize that we do this in some sense over the course of trajectories)

Are the fluctuations so small that we can simply assume that this factor is roughly constant and the estimation over the course of the simulation provides an acceptable stochastic average?