Formation of infinite loops for an interacting bosonic loop soup. (English) Zbl 1534.60141
Summary: We compute the limiting measure for the Feynman loop representation of the Bose gas for a non mean-field energy. As predicted in previous works, for high densities the limiting measure gives positive weight to random interlacements, indicating the quantum Bose-Einstein condensation. We prove that in many cases there is a shift in the critical density compared to the free/mean-field case, and that in these cases the density of the random interlacements has a jump-discontinuity at the critical point.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60F10 | Large deviations |
| 60J65 | Brownian motion |
| 60G50 | Sums of independent random variables; random walks |
References:
| [1] | S. Adams, A Collevecchio, and W. König. A variational formula for the free energy of an interacting many-particle system. Annals of Probability, 39(2):683-728, 2011. MathSciNet: MR2789510 · Zbl 1225.60147 |
| [2] | S. Adams and M. Dickson. An explicit large deviation analysis of the spatial cycle Huang-Yang-Luttinger model. Annales Henri Poincaré, 22(5):1535-1560, 2021. MathSciNet: MR4250742 · Zbl 1481.60063 |
| [3] | S. Adams and M. Dickson. Large deviations analysis for random combinatorial partitions with counter terms. J. Phys. A, 55:255001, 2022. MathSciNet: MR4438639 · Zbl 1507.60037 |
| [4] | I. Armendáriz, P. Ferrari, and S. Yuhjtman. Gaussian random permutation and the boson point process. Communications in Mathematical Physics, 387(3):1515-1547, 2021. MathSciNet: MR4324384 · Zbl 1489.82034 |
| [5] | S. Adams and Q. Vogel. Space-time random walk loop measures. Stochastic Processes and their Applications, 130(4):2086-2126, 2020. MathSciNet: MR4074689 · Zbl 1440.60082 |
| [6] | M. van den Berg, T. Dorlas, J. Lewis, and J. Pulé. The pressure in the Huang-Yang-Luttinger model of an interacting boson gas. Communications in Mathematical Physics, 128(2):231-245, 1990. MathSciNet: MR1043519 · Zbl 0693.60089 |
| [7] | Q. Berger. Notes on random walks in the Cauchy domain of attraction. Probability Theory and Related Fields, 175(1):1-44, 2019. MathSciNet: MR4009704 · Zbl 1479.60086 |
| [8] | N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular Variation. Number 1 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1989. MathSciNet: MR1015093 · Zbl 0667.26003 |
| [9] | Billingsley, P. Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, 1968. MathSciNet: MR0534323 · Zbl 0172.21201 |
| [10] | S. Bose. Plancks Gesetz und Lichtquantenhypothese. Zeitschrift für Physik, 1924. |
| [11] | R. Bahadur and R. Rao. On deviations of the sample mean. The Annals of Mathematical Statistics, 31(4):1015-1027, 1960. MathSciNet: MR0117775 · Zbl 0101.12603 |
| [12] | O. Bratteli and D. Robinson. Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in quantum statistical mechanics. Theoretical and Mathematical Physics. Springer Berlin Heidelberg, 2003. MathSciNet: MR0611508 |
| [13] | V. Betz and D. Ueltschi. Spatial random permutations and infinite cycles. Communications in Mathematical Physics, 285(2):469-501, 2009. MathSciNet: MR2461985 · Zbl 1155.82022 |
| [14] | R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth. On the Lambert W function. Advances in Computational mathematics, 5(1):329-359, 1996. MathSciNet: MR1414285 · Zbl 0863.65008 |
| [15] | F. Comets, S. Popov, and M. Vachkovskaia. Two-dimensional random interlacements and late points for random walks. Communications in Mathematical Physics, 343(1):129-164, 2016. MathSciNet: MR3475663 · Zbl 1336.60185 |
| [16] | A. Drewitz, B. Ráth, and A. Sapozhnikov. An Introduction to Random Interlacements. Springer, 2014. MathSciNet: MR3308116 · Zbl 1304.60008 |
| [17] | A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability. Springer Berlin Heidelberg, 2009. MathSciNet: MR2571413 |
| [18] | A. Einstein. Quantentheorie des einatomigen idealen Gases, 1925. · JFM 51.0755.04 |
| [19] | R. Feynman. Space-time approach to non-relativistic quantum mechanics. Reviews of Modern Physics, 20(2):367-387, 1948. MathSciNet: MR0026940 · Zbl 1371.81126 |
| [20] | R. Feynman. Atomic theory of the \(λ\) transition in helium. Physical Review, 91(6):1291, 1953. · Zbl 0053.48001 |
| [21] | R. Feynman. Statistical Mechanics. Benjamin, Reading, MA, 1972. MathSciNet: MR1634071 |
| [22] | H. Georgii. The equivalence of ensembles for classical systems of particles. Journal of Statistical Physics, 80(5):1341-1378, 1995. MathSciNet: MR1349785 · Zbl 1081.82504 |
| [23] | K. Huang and C. Yang. Quantum-mechanical many-body problem with hard-sphere interaction. Physical Review, 105(3):767, 1957. MathSciNet: MR0083377 · Zbl 0077.20905 |
| [24] | K. Huang, C. Yang, and J. Luttinger. Imperfect Bose gas with hard-sphere interaction. Physical Review, 105(3):776, 1957. MathSciNet: MR0084264 · Zbl 0077.21001 |
| [25] | A. Klenke. Probability Theory: A Comprehensive Course. Universitext. Springer London, 2013. MathSciNet: MR3112259 |
| [26] | J. Lewis. Why do Bosons condense? In Statistical Mechanics and Field Theory: Mathematical Aspects, pages 234-256. Springer, 1986. MathSciNet: MR0862838 · Zbl 0621.76135 |
| [27] | G. Last and M. Penrose. Lectures on the Poisson Process, volume 7. Cambridge University Press, 2017. MathSciNet: MR3791470 |
| [28] | O. Macchi. The coincidence approach to stochastic point processes. Advances in Applied Probability, 7(1):83-122, 1975. MathSciNet: MR0380979 · Zbl 0366.60081 |
| [29] | A. Martin-Löf. A Laplace approximation for sums of independent random variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59(1):101-115, 1982. MathSciNet: MR0643791 · Zbl 0473.60036 |
| [30] | P. Mörters and Y. Peres. Brownian Motion. Cambridge University Press, 2010. MathSciNet: MR2604525 · Zbl 1243.60002 |
| [31] | D. Ruelle. Statistical Mechanics: Rigorous Results. Addison-Wesley, 1969. MathSciNet: MR0289084 · Zbl 0177.57301 |
| [32] | A. Sütő. Percolation transition in the Bose gas. Journal of Physics. A, Mathematical and General, 26(18):4689-4710, 1993. MathSciNet: MR1241339 |
| [33] | A. Sütő. Percolation transition in the Bose gas: II. Journal of Physics A: Mathematical and General, 35(33):6995, 2002. MathSciNet: MR1945163 · Zbl 1066.82006 |
| [34] | A. Sznitman. Vacant set of random interlacements and percolation. Annals of Mathematics, 2010. MathSciNet: MR2680403 |
| [35] | A. Sznitman. On scaling limits and Brownian interlacements. Bulletin of the Brazilian Mathematical Society, New Series, 44(4):555-592, 2013. MathSciNet: MR3167123 · Zbl 1303.60022 |
| [36] | K. Uchiyama. The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge. Proceedings of the London Mathematical Society, 116(3):575-628, 2018. MathSciNet: MR3772617 · Zbl 1391.60204 |
| [37] | D. Ueltschi. Feynman cycles in the Bose gas. Journal of Mathematical Physics, 47(12):123303, 2006. MathSciNet: MR2285149 · Zbl 1112.82028 |
| [38] | Q. Vogel. Emergence of interlacements from the finite volume Bose soup. Stochastic Processes and their Applications, 104227, 2023. MathSciNet: MR4654047 |
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