Applications of character estimates to statistical problems for the symmetric group

Abstract

Let π,σS n be chosen at random. Using character estimates we show that in various aspects the elements πσ i behave like independent random variables. As application we show that almost surely the Cayley graph determined by π and σ has diameter O(n 3 logn), and the directed Cayley-graph has almost surely diameter O(n 4 logn). Further we describe an algorithm for the black-box-recognition of the symmetric group, and show that for each element τ moving a positive proportion of all points, the number of cycles of a random element σ and of τσ are nearly independent.

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Correspondence to Jan-Christoph Schlage-Puchta.

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Schlage-Puchta, JC. Applications of character estimates to statistical problems for the symmetric group. Combinatorica 32, 309–323 (2012). https://doi.org/10.1007/s00493-012-2502-9

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Mathematics Subject Classification (2000)

  • 05C25
  • 20B30
  • 60C05
  • 68W20