Closed form for CLT limit distribution in tensor products of free variables

Derive an explicit closed-form expression for the limiting probability distribution of the normalized sum \bar{x}_N = (x_1 + ··· + x_N − N(λ_1 ··· λ_r))/√N, where x_i = a_1^{(i)} ⊗ ··· ⊗ a_r^{(i)} and, for each s ∈ [r], the elements a_s^{(i)} are identically distributed, free, self-adjoint with mean λ_s and variance σ_s^2. Equivalently, compute the limiting measure corresponding to the free cumulants given by equation (eq-CumulantFreeProd) for general r ≥ 3.

Background

The authors obtain a cumulant-level description of the limit distribution in a central limit theorem for sums of tensor products of free variables, cf. equation (eq-CumulantFreeProd). In the bipartite case (r = 2), they identify the limiting measure as a classical convolution of two dilated semicircular laws followed by a free convolution with a third dilated semicircular law.

Beyond r = 2, although the cumulant formula characterizes the limit, the explicit closed-form distribution is not known. The authors highlight that obtaining a closed expression for the general case remains open.