Work

Central Limit Theorem for tensor products of free variables

Year: 2024

Type: article

Abstract: We establish a Central Limit Theorem for tensor product random variables c Subscript k Baseline colon equals a Subscript k Baseline circled times a Subscript k Baseline $c_k:=a_k \otimes a_k$ c k : = a k ⊗ a k , where left parenthesis a Subscript k Baseline right parenthesis Subscript k element of double struck upper N $(a_k)_{k \in \mathbb {N}}$ ( a k ) k ∈ N is a free family of variables. We show that if the variables a Subscript k $a_k$ a k are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables a Subscript k $a_k$ a k and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

Source: Canadian Journal of Mathematics

DOI: 10.4153/s0008414x24001147

Authors: Cécilia Lancien, Patrick Oliveira Santos, Pierre Youssef