On a random cycle model and its relation to the Bose gas

I will discuss a model of random permutations on a lattice where sites are preferably sent onto neighbours. The main question deals with the possible occurrence of infinite cycles, which should be related to Bose-Einstein condensation.

The random cycle model without interactions behaves very closely to the ideal Bose gas, both qualitatively and quantitatively. This was observed numerically. An interesting problem is to introduce interactions that mimic those of the Bose gas. This may shed light on how the critical temperature depends on the interactions.