  We study the relationship between long cycles and Bose-Einstein condensation
(BEC) in the case of several models. A convenient expression for the density of
particles on cycles of length $q$ is obtained, in terms of $q$ unsymmetrised
particles coupled with a boson field. Using this formulation we reproduce known
results on the Ideal Bose Gas, Mean-Field and Perturbed Mean-Field Models,
where the condensate density exactly equals the long cycle density.
  Then we consider the Infinite-Range-Hopping Bose-Hubbard model, with and
without hard-cores. For the hard-core case, we find we can disregard the
hopping contribution of the q unsymmetrised particles to obtain an exact
expression for the density of particles on long cycles. It is shown that only
the cycle of length one contributes to the cycle density. We conclude that
while the existence of a non-zero long cycle density coincides with the
occurrence of Bose-Einstein condensation, the respective densities are not
necessarily equal.
  For the case of a finite on-site repulsion, we obtain an expression for the
cycle density by neglecting the $q$ unsymmetrised hop. Inspired by the
Approximating Hamiltonian method we conjecture a simplified expression for the
short cycle density as a ratio of single-site partition functions. In the
absence of condensation we prove that this simplification is exact and find
that in this case the long-cycle density vanishes. In the presence of
condensation we justify this simplification when a gauge-symmetry breaking term
is introduced in the Hamiltonian. Assuming our conjecture is correct, we
compare numerically the long-cycle density with the condensate and again find
that though they coexist, in general they are not equal.
