## The Annals of Applied Probability

### Gibbs measures on permutations over one-dimensional discrete point sets

#### Abstract

We consider Gibbs distributions on permutations of a locally finite infinite set $X\subset\mathbb{R}$, where a permutation $\sigma$ of $X$ is assigned (formal) energy $\sum_{x\in X}V(\sigma (x)-x)$. This is motivated by Feynman’s path representation of the quantum Bose gas; the choice $X:=\mathbb{Z}$ and $V(x):=\alpha x^{2}$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 898-929.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355133

Digital Object Identifier
doi:10.1214/14-AAP1013

Mathematical Reviews number (MathSciNet)
MR3313758

Zentralblatt MATH identifier
1314.60042

#### Citation

Biskup, Marek; Richthammer, Thomas. Gibbs measures on permutations over one-dimensional discrete point sets. Ann. Appl. Probab. 25 (2015), no. 2, 898--929. doi:10.1214/14-AAP1013. https://projecteuclid.org/euclid.aoap/1424355133

#### References

• [1] Adams, S., Bru, J.-B. and König, W. (2006). Large deviations for trapped interacting Brownian particles and paths. Ann. Probab. 34 1370–1422.
• [2] Adams, S., Bru, J.-B. and König, W. (2006). Large systems of path-repellent Brownian motions in a trap at positive temperature. Electron. J. Probab. 11 460–485.
• [3] Aizenman, M. and Nachtergaele, B. (1994). Geometric aspects of quantum spin states. Comm. Math. Phys. 164 17–63.
• [4] Berestycki, N. (2006). The hyperbolic geometry of random transpositions. Ann. Probab. 34 429–467.
• [5] Berestycki, N. (2011). Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles. Electron. J. Probab. 16 152–173.
• [6] Berestycki, N. and Durrett, R. (2006). A phase transition in the random transposition random walk. Probab. Theory Related Fields 136 203–233.
• [7] Betz, V. (2013). Random permutations of a regular lattice. Available at arXiv:1309.2955.
• [8] Betz, V. and Ueltschi, D. (2009). Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 469–501.
• [9] Bogolubov, N. (1947). On the theory of superfluidity. Acad. Sci. USSR. J. Phys. 11 23–32.
• [10] Feynman, R. P. (1953). Atomic theory of the $\lambda$ transition in helium. Phys. Rev. 91 1291–1301.
• [11] Fichtner, K.-H. (1991). Random permutations of countable sets. Probab. Theory Related Fields 89 35–60.
• [12] Gandolfo, D., Ruiz, J. and Ueltschi, D. (2007). On a model of random cycles. J. Stat. Phys. 129 663–676.
• [13] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
• [14] Goldschmidt, C., Ueltschi, D. and Windridge, P. (2011). Quantum Heisenberg models and their probabilistic representations. In Entropy and the Quantum II. Contemp. Math. 552 177–224. Amer. Math. Soc., Providence, RI.
• [15] Grosskinsky, S., Lovisolo, A. A. and Ueltschi, D. (2012). Lattice permutations and Poisson–Dirichlet distribution of cycle lengths. J. Stat. Phys. 146 1105–1121.
• [16] Hammond, A. (2012). Infinite cycles in the random stirring model on trees. Available at arXiv:1202.1319.
• [17] Hammond, A. (2012). Sharp phase transition in the random stirring model on trees. Available at arXiv:1202.1322.
• [18] Lieb, E. H., Seiringer, R., Solovej, J. P. and Yngvason, J. (2005). The Mathematics of the Bose Gas and Its Condensation. Oberwolfach Seminars 34. Birkhäuser, Basel.
• [19] Schramm, O. (2005). Compositions of random transpositions. Israel J. Math. 147 221–243.
• [20] Sütő, A. (1993). Percolation transition in the Bose gas. J. Phys. A 26 4689–4710.
• [21] Sütő, A. (2002). Percolation transition in the Bose gas. II. J. Phys. A 35 6995–7002.
• [22] Tóth, B. (1993). Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet. Lett. Math. Phys. 28 75–84.
• [23] Ueltschi, D. (2006). Relation between Feynman cycles and off-diagonal long-range order. Phys. Rev. Lett. 97 170601, 4.
• [24] Zagrebnov, V. A. and Bru, J.-B. (2001). The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350 291–434.