## Electronic Communications in Probability

### The random interchange process on the hypercube

#### Abstract

We prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 4, 9 pp.

Dates
Received: 7 September 2015
Accepted: 7 January 2016
First available in Project Euclid: 3 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1454514624

Digital Object Identifier
doi:10.1214/16-ECP4540

Mathematical Reviews number (MathSciNet)
MR3485373

Zentralblatt MATH identifier
1338.60019

#### Citation

Kotecký, Roman; Miłoś, Piotr; Ueltschi, Daniel. The random interchange process on the hypercube. Electron. Commun. Probab. 21 (2016), paper no. 4, 9 pp. doi:10.1214/16-ECP4540. https://projecteuclid.org/euclid.ecp/1454514624

#### References

• [1] M. Aizenman, B. Nachtergaele, Geometric aspects of quantum spin states, Comm. Math. Phys. 164, 17–63 (1994).
• [2] M. Ajtai, J. Komlos, E. Szemeredi, Largest random component of a k-cube, Combinatorica 2, 1–7 (1982).
• [3] G. Alon, G. Kozma, The probability of long cycles in interchange processes, Duke Math. J. 162, 1567–1585 (2013).
• [4] O. Angel, Random infinite permutations and the cyclic time random walk, Discrete Math. Theor. Comput. Sci. Proc., 9–16 (2003).
• [5] N. Berestycki, Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles, Electr. J. Probab. 16, 152–173 (2011).
• [6] N. Berestycki, G. Kozma, Cycle structure of the interchange process and representation theory, Bull. Soc. Math. France 143, 265–280 (2015).
• [7] J.E. Björnberg, Large cycles in random permutations related to the Heisenberg model, Electr. Commun. Probab. 20, no. 55, 1–11 (2015).
• [8] J.E. Björnberg, The free energy in a class of quantum spin systems and interchange processes, preprint, arXiv:1512.06986 (2015).
• [9] B. Bollobás, Y. Kohayakawa, T. Łuczak, The evolution of random subgraphs of the cube, Random Struct. Algor. 3, 55–90 (1992).
• [10] B. Bollobás, I. Leader, Exact face-isoperimetric inequalities, Europ. J. Combin. 11, 335–340 (1990).
• [11] C. Goldschmidt, D. Ueltschi, P. Windridge, Quantum Heisenberg models and their probabilistic representations, in Entropy and the Quantum II, Contemp. Math. 552, 177–224 (2011); arXiv:1104.0983.
• [12] A. Hammond, Sharp phase transition in the random stirring model on trees, Probab. Theory Rel. Fields 161, 429–448 (2015).
• [13] O. Schramm, Compositions of random transpositions, Isr. J. Math. 147, 221–243 (2005).
• [14] B. Tóth, Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet, Lett. Math. Phys. 28, 75–84 (1993).
• [15] D. Ueltschi, Random loop representations for quantum spin systems, J. Math. Phys. 54, 083301 (2013).