Random Permutations of a Regular Lattice
- 136 Downloads
Spatial random permutations were originally studied due to their connections to Bose–Einstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary conditions, we prove existence of the infinite volume limit under fairly weak assumptions. When the dimension of the lattice is two, we give numerical evidence of a Kosterlitz–Thouless transition, and of long cycles having an almost sure fractal dimension in the scaling limit. Finally we comment on possible connections to Schramm–Löwner curves.
KeywordsRandom permutations Infinite volume limit Kosterlitz–Thouless transition Fractal dimension Schramm–Löwner evolution
Mathematics Subject Classification28A80 60K35 60D05 82B26 82B80
I wish to thank Daniel Ueltschi for introducing me to the topic of SRP, for many good discussions, and for useful comments on the present paper; Thomas Richthammer for letting me have the manuscript  prior to its publication; and Alan Hammond for useful discussions on SLE. Finally I wish to thank one of the anonymous referees of this paper for his/her detailed and insightful comments and suggestions.
- 5.Biskup, M., Richthammer, T.: Gibbs measures on permutations of Z. arXiv:1310.0248 (2013)
- 7.Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics. Cambridge University Press, Cambridge (2000)Google Scholar
- 8.Duminil-Copin, H., Kozma, G., Yadin, A: Supercritical self-avoiding walks are space-filling. arXiv:1110.3074v3 (2012)
- 9.Ellwood, D., Newman, C., Sidoravicius, V., Werner, W.: Probability and Statistical Physics in Two and More Dimensions. AMS Publishing, North Little Rock (2012)Google Scholar
- 10.Falconer, K.: Fractal Geometry, 2nd edn. Wiley, New York (2004)Google Scholar
- 15.Kenyon, R.: Conformal invariance of loops in the double-dimer model. arXiv:1105.4158v2 (2012)
- 19.Lawler, G.F.: Conformally Invariant Processes in the Plane. AMS Publishing, Providence (2005)Google Scholar
- 21.Mermin, N.D., Wagner, H.: Absence of ferromagnetism and antiferromagnetism in one- or two-dimensional Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966); erratum. Phys. Rev. Lett. 17, 1307 (1966)Google Scholar
- 28.Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C.R. Acad. Sci. Paris Ser. I Math. 333(3), 239 (2001)Google Scholar