Simultaneous occurrence of off-diagonal long-range order and infinite permutation cycles in systems of interacting atoms

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.