**Abstract:**
In this talk I consider the relationship between long cycles and
Bose-Einstein condensation (BEC) in the case of several models.
Previous results on the Ideal Bose Gas, Mean-Field and Perturbed
Mean-Field Models indicate that the condensate density exactly equals
the long cycle density. I will consider the infinite-range-hopping
Bose-Hubbard model, with both a finite and an infinite on-site
repulsion interaction, and conclude that while the existence of a
non-zero long cycle density coincides with the occurrence of BEC,
their respective densities are not necessarily equal.