论文标题
非交通概率中功能的Motzkin路径分解
Motzkin path decompositions of functionals in noncommutative probability
论文作者
论文摘要
我们从新的角度研究了自由随机变量的分解。首先，我们表明，相对于归一化的线性功能$φ$的正交复制的混合力矩自然地是用Motzkin单词识别的Motzkin路径来描述的。利用这一事实，我们证明了相对于归一化线性功能的免费产物的自由随机变量的混合力矩是这些变量的正交复制品的混合矩$ n $相对于$φ$的混合矩的总和。该公式的应用之一是分解公式，用于在其布尔累积液中混合随机变量的混合矩，与晶格$ {\ rm nc}（n）$的分解相对应，以sublattices $ \ mathcal {m}（mathcal {m}（m}（m}（w），partitions of partitions of potitionals of nototically of word prodepted prodepted prode po $ $ w $ w $ w $ w $ w $ w。由正交复制品的混合力矩定义的线性函数并由降低的motzkin单词索引起索引的作用，其作用的作用是布尔产品对应于恒定的Motzkin路径的生成空间，而自由产品对应于所有Motzkin路径。
We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional $Φ$ are naturally described in terms of Motzkin paths identified with reduced Motzkin words. Using this fact, we demonstrate that the mixed moments of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of order $n$ of the orthogonal replicas of these variables with respect to $Φ$ with summation extending over the set of reduced Motzkin words of lenght $n$. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice ${\rm NC}(n)$ into sublattices $\mathcal{M}(w)$ of partitions which are monotonically adapted to colors in the word $w$. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.
