Abstract
We consider systems of spatial random permutations, where permutations are weighed according to the point locations. Infinite cycles are present at high densities. The critical density is given by an exact expression. We discuss the relation between the model of spatial permutations and the ideal and interacting quantum Bose gas.
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Aldous D., Diaconis P.: Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. 36, 413–432 (1999)
Arnold P., Moore G.: BEC Transition Temperature of a Dilute Homogeneous Imperfect Bose Gas. Phys. Rev. Lett. 87, 120401 (2001)
Baik J., Deift P., Johannson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, 1119–1178 (1999)
Betz, V., Ueltschi, D.: In preparation
Boland, G., Pulé, J.V.: Long cycles in the infinite-range-hopping Bose-Hubbard model with hard cores. To appear in JSP, available at http://www.ina.otexas.edu/mp_arc/c/08/08-104, 2008
Buffet E., Pulé J.V.: Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24, 1608–1616 (1983)
Davies E.B.: The thermodynamic limit for an imperfect boson gas. Commun. Math. Phys. 28, 69–86 (1972)
Ferrari, P., Prähofer, M., Spohn, H.: Stochastic growth in one dimension and Gaussian multi-matrix models, XIVth International Congress on Mathematical Physics, River Edge, NJ: World Scientific, 2005
Feynman R.P.: Atomic theory of the λ transition in Helium. Phys. Rev. 91, 1291–1301 (1953)
Fichtner K.-H.: Random permutations of countable sets. Probab. Th. Rel. Fields 89, 35–60 (1991)
Gandolfo D., Ruiz J., Ueltschi D.: On a model of random cycles. J. Stat. Phys. 129, 663–676 (2007)
Ginibre, J.: Some applications of functional integration in statistical mechanics. In: “Mécanique statistique et théorie quantique des champs”, Les Houches 1970, C. DeWitt, R. Stora, eds. London: Gorden and Breach, 1971, pp. 327–427
Hainzl C., Seiringer R.: General decomposition of radial functions on \({\mathbb{R}^n}\) and applications to N-body quantum systems. Lett. Math. Phys. 61, 75–84 (2002)
Kashurnikov V.A., Prokof’ev N.V., Svistunov B.V.: Critical temperature shift in weakly interacting Bose gas. Phys. Rev. Lett. 87, 120402 (2001)
Kastening B.: Bose-Einstein condensation temperature of a homogenous weakly interacting Bose gas in variational perturbation theory through seven loops. Phys. Rev. A 69, 043613 (2004)
Kikuchi R.: λ transition of liquid Helium. Phys. Rev. 96, 563–568 (1954)
Kikuchi R., Denman H.H., Schreiber C.L.: Statistical mechanics of liquid He. Phys. Rev. 119, 1823–1831 (1960)
Lebowitz J.L., Lenci M., Spohn H.: Large deviations for ideal quantum systems. J. Math. Phys. 41, 1224–1243 (2000)
Lieb E.H., Seiringer R., Solovej J.P., Yngvason J.: The mathematics of the Bose gas and its condensation, Oberwohlfach Seminars. Birkhäuser, Basel-Boston (2005)
Nho K., Landau D.P.: Bose-Einstein condensation temperature of a homogeneous weakly interacting Bose gas: Path integral Monte Carlo study. Phys. Rev. A 70, 053614 (2004)
Okounkov A.: Random matrices and random permutations. Internat. Math. Res. Notices 20, 1043–1095 (2000)
Okounkov, A.: The uses of random partitions. In Proc. of XIVth International Congress on Mathematical Physics, River Edge, NJ: World Scientific, 2005, pp. 379–403
Penrose O., Onsager L.: Bose-Einstein condensation and liquid Helium. Phys. Rev. 104, 576 (1956)
Pollock E.L., Ceperley D.M.: Path-integral computation of superfluid densities. Phys. Rev. B 36, 8343–8352 (1987)
Sütő A.: Percolation transition in the Bose gas. J. Phys. A 26, 4689–4710 (1993)
Sütő A.: Percolation transition in the Bose gas II. J. Phys. A 35, 6995–7002 (2002)
Ueltschi D.: Feynman cycles in the Bose gas. J. Math. Phys. 47, 123302 (2006)
Ueltschi D.: Relation between Feynman cycles and off-diagonal long-range order. Phys. Rev. Lett. 97, 170601 (2006)
Ueltschi, D.: The model of interacting spatial permutations and its relation to the Bose gas. http://arxiv.org.abs/0712.2443, v3 [cond-mat, stat-mech], 2007
Zagrebnov V.A., Bru J.-B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Reports 350, 291–434 (2001)
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Betz, V., Ueltschi, D. Spatial Random Permutations and Infinite Cycles. Commun. Math. Phys. 285, 469–501 (2009). https://doi.org/10.1007/s00220-008-0584-4
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DOI: https://doi.org/10.1007/s00220-008-0584-4