Infinite Cycles and Bose–Einstein Condensation

Definitions and mathematical statement

The supplied record concerns an interacting spatial random-permutation model in integer dimension $d\geq 3$, with parameters $\beta$ and $\rho$, and two variational quantities denoted by $\chi^{(\leq)}(\beta,\rho)$ and $\chi^{(=)}(\beta,\rho)$. Let $P$ denote an optimizer of $\chi^{(\leq)}(\beta,\rho)$, whenever such an optimizer exists. These symbols retain their source meanings: the record does not specify the interaction, admissible parameter ranges, variational functional, constraint sets, or mathematical definitions of cycle loss of mass and Bose--Einstein condensation.

Source-level question. For this model, are the following three conditions equivalent?

1. There is loss of probability mass into infinitely long cycles for the optimizer $P$.
2. The variational values satisfy \[ \chi^{(\leq)}(\beta,\rho)\neq\chi^{(=)}(\beta,\rho). \]
3. Bose--Einstein condensation occurs.

For each fixed admissible $\beta$, is such loss absent for all sufficiently small admissible densities $\rho$, and present for all sufficiently large admissible densities $\rho$?

Unresolved qualifications. The phrase "an optimizer" does not determine whether the question concerns every optimizer, at least one optimizer, or a selected optimizer. Accordingly, neither an optimizer quantifier nor optimizer independence is asserted here. The record also does not specify the thermodynamic limit, boundary conditions, interaction hypotheses, the measure whose mass is lost, the order of limits defining infinitely long cycles, or the criterion for Bose--Einstein condensation. The constraint meanings suggested by the superscripts $(\leq)$ and $(=)$ are not defined in the supplied text. Thus this is a statement-only transcription of the proposed equivalence and density-regime question; a fully specified mathematical formulation requires those source definitions and hypotheses. No conclusion at a transition density is supplied.

Short English statement

The question asks whether, in dimension $d\geq3$, mass lost into infinitely long permutation cycles, inequality of the two stated variational values, and Bose--Einstein condensation are equivalent. It also asks whether cycle mass loss is absent at sufficiently small density and present at sufficiently large density for each fixed $\beta$. The model definitions and the intended quantifier over optimizers remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30001480$, title Infinite Cycles and Bose--Einstein Condensation, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The record lists proposed year 2010 and supplies no named proposer.
• Supplied source citation: Phase Transitions (2010), Oberwolfach Reports, DOI 10.4171/owr/2010/24, pp. 1431--1484. https://doi.org/10.4171/owr/2010/24. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled Adams--Collevecchio--König: https://doi.org/10.1214/10-AOP565. No title, theorem reference, or specific supporting passage is supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and states that variational theory establishes important regimes for the interacting particle system, while the conjectural equivalence between cycle loss of mass and Bose--Einstein condensation remains open. Its listed sources checked are OWR 2010/24 and Adams--Collevecchio--König, linked above. This is supplied dated curation derived from the OWR curation, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry supplies statement status corrected_verified and reports that the mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification assertion is part of the supplied record, not an independent verification performed for this draft.