Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique

Sandrine Roussel

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 111-135
  • ISSN: 0010-1354


The main purpose of this paper is to exhibit the cutoff phenomenon, studied by Aldous and Diaconis [AD]. Let Q * k denote a transition kernel after k steps and π be a stationary measure. We have to find a critical value k n for which the total variation norm between Q * k and π stays very close to 1 for k k n , and falls rapidly to a value close to 0 for k k n with a fall-off phase much shorter than k n . According to the work of Diaconis and Shahshahani [DS], one can naturally conjecture, for a conjugacy class with n-c fixed points, with c n , that the associated random walk presents a cutoff, with critical value equal to (1/c)nln(n). Using Fourier analysis, we prove that, in this context, the critical value can not be less than (1/c)nln(n). We also prove that the conjecture is true for conjugacy classes with at least n-6 fixed points and for a conjugacy class of cycle length 7.

How to cite


Roussel, Sandrine. "Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique." Colloquium Mathematicae 86.1 (2000): 111-135. <>.

author = {Roussel, Sandrine},
journal = {Colloquium Mathematicae},
language = {fre},
number = {1},
pages = {111-135},
title = {Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique},
url = {},
volume = {86},
year = {2000},

AU - Roussel, Sandrine
TI - Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 111
EP - 135
LA - fre
UR -
ER -


  1. [AD] D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. Appl. Math. 8 (1987), 69-97. 
  2. [Ay] R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., Providence, RI, 1963. 
  3. [D1] P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 1659-1664. Zbl0849.60070
  4. [D2] P. Diaconis, Group Representations in Probability and Statistics, IMS Lecture Notes Monogr. Ser. 11, Hayward, CA, 1988. 
  5. [DS] P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), 159-179. Zbl0485.60006
  6. [Fel] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd ed., Wiley, New York, 1968. 
  7. [Ing] R. E. Ingram, Some characters of the symmetric group, Proc. Amer. Math. Soc. 1 (1950), 358-369. Zbl0054.01103
  8. [Jam] G. D. James, The Representation Theory of the Symmetric Group, Lecture Notes in Math. 682, Springer, Berlin, 1978. 
  9. [JK] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981. 
  10. [R1] Y. Roichman, Upper bound on the characters of the symmetric groups, Invent. Math. 125 (1996), 451-485. Zbl0854.20015
  11. [R2] S. Roussel, Marches aléatoires sur le groupe symétrique, thèse de doctorat (en préparation), 1999. 
  12. [Sag] B. E. Sagan, The Symmetric Group , Representations , Combinatorial Algorithms and Symmetric Functions, Wadsworth and Brooks/Cole Math. Ser., 1991. 
  13. [SC1] L. Saloff-Coste, Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z. 217 (1994), 641-677. Zbl0815.60074
  14. [SC2] L. Saloff-Coste, Lectures on finite Markov chains, in: Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1665, Springer, 1997, 301-413. Zbl0885.60061
  15. [Ser] J. P. Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1977. 

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