改进的Thorp混洗和L-反转链的混合时间界
Improved mixing time bounds for the Thorp shuffle and L-reversal chain
摘要：
我们证明了一个定理，减少绑定的混合时间的洗牌，以验证一个条件，只涉及对卡，然后我们用它来获得改进的界限为两个先前研究的模型。E.索普在1973年引入了以下洗牌模型。假设牌的数目n是偶数。把甲板切成两等份。根据掷硬币的结果，从左边或右边的牌堆中丢出第一张牌。然后从另一堆上掉下来。继续这样做，直到两堆都空了。我们得到了一个O（log^4 n）的混合时间界。以前，最著名的界限是O（log^{29} n），以前的证明只对n的2次幂有效。我们还分析了下面的模型，称为L-反转链，由Durrett介绍。有n张牌排成一圈。每一步，长度最多为L的卡片的间隔被随机地均匀地选择，并且其顺序被颠倒。时间复杂度为O（max（n，n^3/L^3）log n）。我们得到了一个在其因子O（log^2 n）内的界，这是该猜想在多对数因子内的第一个界。
英文摘要：
We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from 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 obtain a mixing time bound of O(log^4 n). Previously, the best known bound was O(log^{29} n) and previous proofs were only valid for n a power of 2. We also analyze the following model, called the L-reversal chain, introduced by Durrett. There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/L^3) log n). We obtain a bound that is within a factor O(log^2 n) of this,the first bound within a poly log factor of the conjecture.
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