Commutators in finite free probability, I

自由概率论中的换位子和

Sums of Commutators in Free Probability

本文研究自由随机变量换位子的线性张成，证明其是唯一满足保持自由无穷可分性等三个等价性质的二次型，并利用非交叉划分上的对合作为主要组合工具。

关于含半圆变量的自由交换子的若干性质

On some properties of free commutators with semicircular variables

本文针对含半圆元的自由变量交换子展开研究，证明其与半圆元的和满足自由加法卷积恒等式，且当x自由无穷可分时对应多项式也具备该性质。

The probability of generating the symmetric group with a commutator condition

Let B(n) be the set of pairs of permutations from the symmetric group of degree n with a 3-cycle commutator, and let A(n) be the set of those pairs which generate the symmetric or the alternating group of degree n. We find effective formulas for calculating the cardinalities of both sets. More precisely, we show that #B(n)/n! is a discrete convolution of the partition function and a linear combination of divisor functions, while #A(n)/n! is the product of a polynomial and Jordan's totient function. In particular, it follows that the probability that a pair of random permutations with a 3-cycle…

二阶自由累积量：乘积、交换子与反对易子

Second order free cumulants: product, commutator, and anti-commutator

本文研究两个二阶自由独立随机变量的乘积、交换子与反对易子，给出其二阶自由累积量的一般求和公式，并应用于二阶自由半圆变量。

Conjugacy growth of commutators

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively, $G_1 * G_2$) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of $G_1$ and $G_2$). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy cla…

关于两个自由随机变量的反对易子

On the anti-commutator of two free random variables

本文利用二部图和非交叉划分给出两个自由独立随机变量反对易子自由累积量的计算公式，并借助仙人掌图推广到k个自由随机变量的二次型。