Borel-Tanner Distribution
المؤلف:
Berestycki, N
المصدر:
"The Hyperbolic Geometry of Random Transpositions."
الجزء والصفحة:
...
23-3-2021
2019
Borel-Tanner Distribution
Let 
be the set of permutations of ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/Borel-TannerDistribution/Inline2.gif" style="height:15px; width:5px" />1, 2, ..., 
![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/Borel-TannerDistribution/Inline4.gif" style="height:15px; width:5px" />, 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).
REFERENCES:
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. Fields 136, 203-233, 2006.
Haight, F. A. and Breuer, M. A. "The Borel-Tanner Distribution." Biometrika 47, 143-150, 1960.
Trott, M. The Mathematica GuideBook for Numerics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/.
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