Let
be the set of permutations of 1, 2, ..., , and let be the continuous time random walk on that results when randomly chosen transpositions are performed
at rate 1. Let be the distance from the identity at time , i.e., the minimum number of transpositions needed to return
to .
Then as ,
,
where
(Berestycki 2004; Berestycki and Durrett 2004), where is known as the Borel-Tanner distribution (Trott 2006,
p. 284).
The Borel-Tanner distribution for complex is plotted above in the complex plane (Trott 2006, p. 284).
Interestingly, this function has the value for (Berestycki 2004; Trott 2006, p. 284).
Berestycki, N. "The Hyperbolic Geometry of Random Transpositions." 31 Oct 2004. http://arxiv.org/abs/math.PR/0411011.Berestycki,
N. and Durrett, R. "A Phase Transition in the Random Transposition Random Walk."
Probab. Theor. Rel. Fields136, 203-233, 2006.Haight,
F. A. and Breuer, M. A. "The Borel-Tanner Distribution." Biometrika47,
143-150, 1960.Trott, M. The
Mathematica GuideBook for Numerics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/.
be the set of permutations of 
1, 2, ..., 
, and let 
be the continuous time random walk on 
that results when randomly chosen transpositions are performed
at rate 1. Let 
be the distance from the identity 
at time 
, i.e., the minimum number of transpositions needed to return
to 
.
Then as 
,
,
where
is known as the Borel-Tanner distribution (Trott 2006,
p. 284).
is plotted above in the complex plane (Trott 2006, p. 284).
for 
(Berestycki 2004; Trott 2006, p. 284).
