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The Mathematics of MagicVariation Distances and the Cutoff Phenomenon in Card Shuffling
Megan Poole
Senior TalkFebruary 24, 2015
Megan Poole Card Shuffling
Types of Shuffles
Perfect Shuffle
Perfect interleaving of a deck.
Top-in-at-Random-Shuffle
Top card inserted into the deck at random.
Riffle Shuffle
Random interleaving of two halves of a deck.
Megan Poole Card Shuffling
To begin...
We must know:
Deck order
Deck size
Type of Shuffle
Measure of Randomness
Megan Poole Card Shuffling
Variation Distances and the Cutoff Phenomenon
Prove:
d(n log n + cn) ≤ e−c ; all c ≥ 0, n ≥ 2
d(n log n − cnn)→ 1 as n→∞; all cn →∞
Megan Poole Card Shuffling
Variation Distances and the Cutoff Phenomenon
Prove:
d(n log n + cn) ≤ e−c ; all c ≥ 0, n ≥ 2
d(n log n − cnn)→ 1 as n→∞; all cn →∞
Megan Poole Card Shuffling
Coupon Collector’s Problem
Problem
Given n coupons, how many coupons do you expect you need todraw with replacement before having drawn each coupon at leastonce?
Megan Poole Card Shuffling
Proving Theorem 1
Prove:
d(n log n + cn) ≤ e−c ; all c ≥ 0, n ≥ 2
d(n log n − cnn)→ 1 as n→∞; all cn →∞
Megan Poole Card Shuffling
Basic Math of Card Shuffling
GSR Model of Shuffling
Cut: Binomial Distribution
Shuffle: Probability determined by number of cards in eachhalf of the cut
Cutoff Phenomenon: 3/2 log2 n
Rising Sequences
Megan Poole Card Shuffling
References
M. Aigner & G. Ziegler: Proofs from the book, (4th ed.). Berlin:Springer. (2010), 185-194.
D. Aldous & P. Diaconis: Shuffling cards and stopping times,Amer. Math. Monthly 93 (1986), 333-348.
D. Bayer & P. Diaconis: Trailing the dovetail shuffle to its lair,Annals Applied Probability 2 (1992), 294-313.
Megan Poole Card Shuffling