18,444 research outputs found
Partial mixing of semi-random transposition shuffles
We show that for any semi-random transposition shuffle on cards, the
mixing time of any given cards is at most , provided . In the case of the top-to-random transposition shuffle we show that
there is cutoff at this time with a window of size O(n), provided further that
as (and no cutoff otherwise). For the
random-to-random transposition shuffle we show cutoff at time
for the same conditions on . Finally, we analyse the cyclic-to-random
transposition shuffle and show partial mixing occurs at time for some just larger than 1/2. We prove these results by relating
the mixing time of cards to the mixing of one card. Our results rely
heavily on coupling arguments to bound the total variation distance.Comment: 23 pages, 4 figure
Cutoff for the cyclic adjacent transposition shuffle
We study the cyclic adjacent transposition (CAT) shuffle of cards, which
is a systematic scan version of the random adjacent transposition (AT) card
shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at
, which concludes that it is twice as fast as the
AT shuffle.Comment: 26 pages, 3 figure
The shuffle block dynamics
We introduce and analyze the shuffle on cards, a natural
generalization of the celebrated random adjacent transposition shuffle. In the
shuffle, we choose uniformly at random a block of consecutive cards,
and shuffle these cards according to a permutation chosen uniformly at random
from the symmetric group on elements. We study the total-variation mixing
time of the shuffle when the number of cards goes to infinity,
allowing also to grow with . In particular, we show that the cutoff
phenomenon occurs when .Comment: 21 pages, 1 figur
The mixing time of the Thorp shuffle
The Thorp shuffle is defined as follows. Cut the deck into two equal piles.
Drop the first card from the left pile or the right pile according to the
outcome of a fair coin flip; then drop from the other pile. Continue this way
until both piles are empty. We show that the mixing time for the Thorp shuffle
with cards is polynomial in .Comment: 21 page
Cyclotomic shuffles
Analogues of 1-shuffle elements for complex reflection groups of type
are introduced. A geometric interpretation for in terms
of rotational permutations of polygonal cards is given. We compute the
eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra
of the group . Considering shuffling as a random walk on the group
, we estimate the rate of convergence to randomness of the
corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue
in the cyclotomic Hecke algebra for and small
A complete classification of shuffle groups
For positive integers and , the shuffle group is generated
by the permutations of a deck of cards performed by cutting the deck
into piles with cards in each pile, and then perfectly interleaving
these cards following certain order of the piles. For , the shuffle
group was determined by Diaconis, Graham and Kantor in 1983. The
Shuffle Group Conjecture states that, for general , the shuffle group
contains whenever and is not
a power of . In particular, the conjecture in the case was posed by
Medvedoff and Morrison in 1987. The only values of for which the Shuffle
Group Conjecture has been confirmed so far are powers of , due to recent
work of Amarra, Morgan and Praeger based on Classification of Finite Simple
Groups. In this paper, we confirm the Shuffle Group Conjecture for all cases
using results on -transitive groups and elements of large fixed point ratio
in primitive groups
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