The Mixing Time for a Random Walk on the Symmetric Group Generated by Random Involutions

IF 0.7 4区 数学
Megan Bernstein
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引用次数: 8

Abstract

International audience The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.
随机对合生成的对称群上随机漫步的混合时间
对合行走是从参数p的二项分布中采样若干个2循环的对合而产生的对称群上的随机行走。这是对称群上的惰性转置行走的并行化。本文显示的对合步数为1≤p≤1固定,在log1/p(n)步和log2/(1+p)(n)步之间足够大2。本文介绍了一种利用对称群的表示特征的特征多项式求共轭类生成的对称群上随机游动特征值的新方法。这在计算大特征值时特别有效。较小的特征值是通过发展单调关系来处理的,这种单调关系在足够的时间后也给出了似然顺序,即从最可能状态到最不可能状态的顺序。从后孔信息论出发,引入了随机行走的概念,研究了酉群上随机行走的一个猜想。
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来源期刊
自引率
14.30%
发文量
39
期刊介绍: DMTCS is a open access scientic journal that is online since 1998. We are member of the Free Journal Network. Sections of DMTCS Analysis of Algorithms Automata, Logic and Semantics Combinatorics Discrete Algorithms Distributed Computing and Networking Graph Theory.
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