Long cycles and Bose-Einstein condensation in the infinite-range-hopping Bose-Hubbard model

We study the relation between long cycles and Bose-Condensation in the Infinite range Bose-Hubbard model with a hard core. We calculate the density of particles on long cycles in the thermodynamic limit and find that the existence of a non-zero long cycle density coincides with the occurence of Bose-Einstein condensation but this density is not equal to that of the Bose condensate. We give some partial results for the Infinite range Bose-Hubbard model without a hard core.