## Duke Mathematical Journal

### The probability of long cycles in interchange processes

#### Abstract

We examine the number of cycles of length $k$ in a permutation as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows us to study the formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time $t$, and estimates for the cases of shorter cycles.

#### Article information

Source
Duke Math. J., Volume 162, Number 9 (2013), 1567-1585.

Dates
First available in Project Euclid: 11 June 2013

https://projecteuclid.org/euclid.dmj/1370955539

Digital Object Identifier
doi:10.1215/00127094-2266018

Mathematical Reviews number (MathSciNet)
MR3079255

Zentralblatt MATH identifier
1269.82041

#### Citation

Alon, Gil; Kozma, Gady. The probability of long cycles in interchange processes. Duke Math. J. 162 (2013), no. 9, 1567--1585. doi:10.1215/00127094-2266018. https://projecteuclid.org/euclid.dmj/1370955539

#### References

• [1] G. Alon and G. Kozma, Ordering the representations of $S_{n}$ using the interchange process, Canad. Math. Bull. 56 (2013), 13–30.
• [2] O. Angel, “Random infinite permutations and the cyclic time random walk” in Discrete Random Walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003, 9–16.
• [3] R. Bacher, Valeur propre minimale du laplacien de Coxeter pour le groupe symétrique, J. Algebra 167 (1994), 460–472.
• [4] N. Berestycki, Emergence of giant cycles and slowdown transition in random transpositions and k-cycles, Electron. J. Probab. 16 (2011), 152–173.
• [5] N. Berestycki and R. Durrett, A phase transition in the random transposition random walk, Probab. Theory Related Fields 136 (2006), 203–233.
• [6] N. Berestycki and G. Kozma, Cycle structure of the interchange process and representation theory, preprint, arXiv:1205.4753v1 [math.PR].
• [7] P. Caputo, T. M. Liggett, and T. Richthammer, Proof of Aldous’ spectral gap conjecture, J. Amer. Math. Soc. 23 (2010), 831–851.
• [8] P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), 159–179.
• [9] N. Eriksen and A. Hultman, Estimating the expected reversal distance after a fixed number of reversals, Adv. in Appl. Math. 32 (2004), 439–453.
• [10] W. Fulton and J. Harris, Representation Theory: A First Course, Grad. Texts in Math. 129, Springer, New York, 1991.
• [11] A. Hammond, Infinite cycles in the random stirring model on trees, preprint, arXiv:1202.1319v2 [math.PR].
• [12] Alan Hammond, Sharp phase transition in the random stirring model on trees, preprint, arXiv:1202.1322v2 [math.PR].
• [13] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, Mass., 1981.
• [14] D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times, Amer. Math. Soc., Providence, 2009.
• [15] E. Lubetzky and A. Sly, Explicit expanders with cutoff phenomena, Electron. J. Probab. 16 (2011), 419–436.
• [16] R. Montenegro and P. Tetali, Mathematical Aspects of Mixing Times in Markov Chains, Found. Trends Theor. Comput. Sci. 1, NOW, Boston, 2006.
• [17] B. Morris, The mixing time for simple exclusion, Ann. Appl. Probab. 16 (2006), 615–635.
• [18] R. I. Oliveira, Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk, to appear in Ann. Probab., preprint, arXiv:1007.2669v3 [math.PR].
• [19] R. Pemantle, A shuffle that mixes sets of any fixed size much faster than it mixes the whole deck, Random Structures Algorithms 5 (1994), 609–626.
• [20] B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed., Grad. Texts in Math. 203, Springer, New York, 2001.
• [21] O. Schramm, Compositions of random transpositions, Israel J. Math. 147 (2005), 221–243.
• [22] B. Tóth, Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet, Lett. Math. Phys. 28 (1993), 75–84.
• [23] N. V. Tsilevich, Spectral properties of the periodic Coxeter Laplacian in the two-row ferromagnetic case (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 378 (2010), 111-132; English translation in J. Math. Sci. (N. Y.) 174 (2011), 58–70.
• [24] D. B. Wilson, Mixing times of Lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab. 14 (2004), 274–325.