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Information Loss in Riffle Shuffling

Published online by Cambridge University Press:  14 February 2002

DUDLEY STARK ,
A. GANESH  and
NEIL O’CONNELL
DUDLEY STARK
Affiliation:
Queen Mary, University of London, Mile End Road, London E1 4NS, UK (e-mail: [email protected])
A. GANESH
Affiliation:
Microsoft Research, St. George House, 1 Guildhall Street, Cambridge CB2 3NH, UK (e-mail: [email protected] )
NEIL O’CONNELL
Affiliation:
BRIMS, Hewlett-Packard Laboratories, Filton Road, Bristol BS34 8QZ, UK (e-mail: [email protected])
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Abstract

We study the asymptotic behaviour of the relative entropy (to stationarity) for a commonly used model for riffle shuffling a deck of n cards m times. Our results establish and were motivated by a prediction in a recent numerical study of Trefethen and Trefethen. Loosely speaking, the relative entropy decays approximately linearly (in m) for m < log2n, and approximately exponentially for m > log2n. The deck becomes random in this information-theoretic sense after m = 3/2 log2n shuffles.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 1 , January 2002 , pp. 79 - 95
Copyright
2002 Cambridge University Press

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