Cumulants, free cumulants and half-shuffles
Cumulants, free cumulants and half-shuffles
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累积量、自由累积量和半洗牌
DOI：
10.1098/rspa.2014.0843
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
F. Patras
中科院分区：
文献类型：
--
作者：
K. Ebrahimi;F. Patras
在自由概率论中，自由累积量是作为经典累积量的适当模拟而引入的。有一个混合的相似性和差异，当考虑两族累积量。而经典累积量的组合学是很好地表示在集合分区，自由累积量的描述，并经常介绍在非交叉集合分区。经典累积量和自由累积量的形式级数方法也有很大的不同。本研究的目的是对这些现象提出一种不同的方法。也就是说，我们表明，累积量，无论是经典的或自由的，可以理解的代数和组合学的基本交换以及非交换（半）洗牌和（半）unshuffles。作为推论，累积量和自由累积量可以通过线性不动点方程来刻画。我们研究了这些线性不动点方程的指数解，它们分别具有经典累积量和自由累积量的交换性和非交换性。
Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differs. The purpose of this study is to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-) unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical and free cumulants.