Card shuffling and Diophantine approximation
نویسندگان
چکیده
منابع مشابه
Card Shuffling and Diophantine Approximation
The “overlapping-cycles shuffle” mixes a deck of n cards by moving either the nth card or the (n − k)th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of k and n, has surprising behavior. For example, suppose k is the closest integer to αn for a fixed real α ∈ (0, 1). Then for rational α the spectral...
متن کامل3 . 3 Card Shuffling
Given a deck of n cards, how many times must we shuffle it to make it “random”? Of course, the answer depends upon the method of shuffling which is used and what we mean by “random.” We shall begin the study of this question by considering a standard model for the riffle shuffle. We begin with a deck of n cards, which we will assume are labelled in increasing order with the integers from 1 to n...
متن کاملDiophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملOn Transience of Card Shuffling
We present simple proofs of transience/recurrence for certain card shuffling models, that is, random walks on the infinite symmetric group. 1. Card shuffling models In this note, we consider several models of shuffling an infinite deck of cards. One of these models has been considered previously by Lawler [La]; our methods (using flows, shorting and comparison of Dirichlet forms) will – partial...
متن کاملJuggling and Card Shuffling Meet Mathematical Fonts
We explore two of Ron Graham’s passions—juggling patterns and perfect card shuffling— through one of our passions, mathematical fonts. First, for each letter of the English alphabet, we design a one-person three-ball juggling pattern where the balls trace out the letter (possibly rotated 90◦). Second, using a deck of 26 cards labeled A through Z, we show how to perform a sequence of in/out perf...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Annals of Applied Probability
سال: 2008
ISSN: 1050-5164
DOI: 10.1214/07-aap484