Percolation transition in the Bose gas II

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.

The paper is completed with the proof of Eq. (34) and with a remark after Eq. (44) on the existence for any m>ln(rho/rho_0) of an infinite cycle which contains a fraction between exp{-(m+1)} and exp{-m} of the total number of particles

Cited by in corpus (36)

• Spatial random permutations and infinite cycles
• Spatial random permutations and Poisson-Dirichlet law of cycle lengths
• Spatial random permutations with small cycle weights
• Lattice permutations and Poisson-Dirichlet distribution of cycle lengths
• On a model of random cycles
• Long Cycles in a Perturbed Mean Field Model of a Boson Gas
• Fluctuations of the Bose-Einstein condensate
• A variational formula for the free energy of an interacting many-particle system
• Feynman cycles in the Bose gas
• Critical Temperature of Dilute Bose Gases
• The relation between Feynman cycles and off-diagonal long-range order
• Infinite cycles in the random stirring model on trees
• Random permutations of a regular lattice
• Random partitions in statistical mechanics
• A path-integral analysis of interacting Bose gases and loop gases
• Macroscopic loops in the Bose gas, Spin O(N) and related models
• Off-diagonal long-range order, cycle probabilities, and condensate fraction in the ideal Bose gas
• Scaling approach to existence of long cycles in Casimir boxes
• Formation of infinite loops for an interacting bosonic loop soup
• Gibbs measures on permutations over one-dimensional discrete point sets
• Shift in critical temperature for random spatial permutations with cycle weights
• Long Cycles in the Infinite-Range-Hopping Bose-Hubbard Model with Hard Cores
• Large deviation analysis for classes of interacting Bosonic cycle counts
• Large deviations analysis for random combinatorial partitions with counter terms
• Emergence of interlacements from the finite volume Bose soup
• Large deviations for empirical path measures in cycles of integer partitions
• Finite cycle Gibbs measures on permutations of ${\mathbb{Z}}^d$
• The model of interacting spatial permutations and its relation to the Bose gas
• Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
• Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula
• Infinite cycles of interacting bosons
• Localised pair formation in bosonic flat-band Hubbard models
• Long Cycles in the Infinite-Range-Hopping Bose-Hubbard Model
• Large deviations for many Brownian bridges with symmetrised initial-terminal condition
• Infinite Cycles in Boson Lattice Models
• Simultaneous occurrence of off-diagonal long-range order and infinite permutation cycles in systems of interacting atoms