Percolation transition in the Bose gas II

TL;DR

This paper proves that cycle percolation in the Bose gas coincides with Bose-Einstein condensation, demonstrating the presence of infinite macroscopic cycles in the condensate for both perfect and mean-field models.

Contribution

It provides a complete proof linking cycle percolation to Bose-Einstein condensation in perfect and imperfect Bose gases, confirming the conjecture from prior work.

Findings

Cycle percolation occurs if and only if Bose-Einstein condensation occurs.

In the condensate, there are infinitely many macroscopic cycles.

The proof applies to both perfect and mean-field Bose gases.

Abstract

In an earlier paper (J. Phys. A: Math. Gen. 26 (1993) 4689) we introduced the notion of cycle percolation in the Bose gas and conjectured that it occurs if and only if there is Bose-Einstein condensation. Here we give a complete proof of this statement for the perfect and the imperfect (mean-field) Bose gas and also show that in the condensate there is an infinite number of macroscopic cycles.