Transition Temperature of the homogeneous, weakly interacting Bose gas

We present a Monte Carlo calculation for up to $N\sim 20000$ bosons in 3 D to determine the shift of the transition temperature due to small interactions $a$. We generate independent configurations of the ideal gas. At finite $N$, the superfluid density changes by a certain correlation function in the limit $a\to 0$; the $N\to \mathrm{\infty }$ limit is taken afterwards. We argue that our result is independent of the order of limits. Detailed knowledge of the non-interacting system for finite $N$ allows us to avoid finite-size scaling assumptions.