Condensation phenomena in particle systems typically occur as one of two distinct types: either as a spontaneous symmetry breaking in a homogeneous system, in which particle interactions enforce condensation in a randomly located site, or as an explicit symmetry breaking in a system with background disorder, in which particles condensate in the site of extremal disorder. In this paper we confirm a recent conjecture by Godreche and Luck by showing, for a zero range process with weak site disorder, that there exists a phase where condensation occurs with an intermediate type of symmetry-breaking, in which particles condensate in a site randomly chosen from a range of sites favoured by disorder. We show that this type of condensation is characterised by the occurrence of a Gamma distribution in the law of the disorder at the condensation site. We further investigate fluctuations of the condensate size and confirm a phase diagram, again conjectured by Godreche and Luck, showing the existence of phases with normal and anomalous fluctuations.
Mailler, C., Morters, P., & Ueltschi, D. (2016). Condensation and symmetry-breaking in the zero-range process with weak site disorder. Stochastic Processes and their Applications, 126(11), 3283-3309. https://doi.org/10.1016/j.spa.2016.04.028