Off-diagonal long-range order, cycle probabilities, and condensate fraction in the ideal Bose gas

We discuss the relationship between the cycle probabilities in the path-integral representation of the ideal Bose gas, off-diagonal long-range order, and Bose--Einstein condensation. Starting from the Landsberg recursion relation for the canonic partition function, we use elementary considerations to show that in a box of size L^3 the sum of the cycle probabilities of length k >> L^2 equals the off-diagonal long-range order parameter in the thermodynamic limit. For arbitrary systems of ideal bosons, the integer derivative of the cycle probabilities is related to the probability of condensing k bosons. We use this relation to derive the precise form of the \pi_k in the thermodynamic limit. We also determine the function \pi_k for arbitrary systems. Furthermore we use the cycle probabilities to compute the probability distribution of the maximum-length cycles both at T=0, where the ideal Bose gas reduces to the study of random permutations, and at finite temperature. We close with comments on the cycle probabilities in interacting Bose gases.