  Based on the paper "Fourier formula for quantum partition functions",
arXiv:2106.10032 [math-ph], we show that in an infinite system of identical
bosons interacting via a positive-type pair potential there is off-diagonal
long-range order if and only if a nonzero fraction of the particles form
infinite permutation cycles. In particular, there is Bose-Einstein condensation
if and only if the diverging cycle lengths increase at least as fast with $N$,
the number of particles, as $N^{2/d}$ in $d\geq 3$ dimensions. This extends a
similar result known for the ideal Bose gas.
