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Achievability of the Connes-embeddable free Wasserstein distance by invariant random multi-matrix couplings

Determine whether, for limiting non-commutative laws (or types) μ and ν and the Connes-embeddable Biane–Voiculescu–Wasserstein distance d_{W,CEP}(μ,ν), there exist invariant random multi-matrix ensembles X^{(n)} and Y^{(n)} of the form dμ^{(n)}(X) ∝ e^{-n^2 V^{(n)}(X)} dX and dν^{(n)}(Y) ∝ e^{-n^2 W^{(n)}(Y)} dY (with V^{(n)}, W^{(n)} given by non-commutative *-polynomial data) such that a coupling of (X^{(n)}, Y^{(n)}) asymptotically attains d_{W,CEP}(μ,ν).

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Background

The paper studies how free entropy and Wasserstein distance interact in non-commutative settings and their large-n matrix limits. For limiting objects μ and ν, the Biane–Voiculescu metric restricted to Connes-embeddable algebras, d_{W,CEP}, is the relevant free analog of classical Wasserstein distance.

A central issue is whether an optimal free transport can be realized by actual couplings of invariant matrix ensembles generated by potentials as in equation (1.1). Invariance forces the mass of X{(n)} to distribute approximately uniformly over microstate spaces, and any construction of an optimal coupling must reconcile pointwise minimality with this uniformity property for Y{(n)} as well.

The authors highlight that these constraints may conflict and provide a counterexample (for laws) showing that one cannot simultaneously achieve both the "correct" entropy and the free Wasserstein distance. But the general question of achieving the distance with invariant couplings, independent of entropy matching, remains unresolved.

References

However, we do not know if this distance can be achieved by a coupling of random multi-matrix models X{(n)} and Y{(n)} given as in (1.1). For this to happen, the minimal distance has to be achievable by some value of Y{(n)} for most given values of X{(n)} with the same limiting law, i.e., most choices of X in Voiculescu's microstate space Γ_R{(n)} associated to neighborhoods of μ, since invariance means that the probability mass of X{(n)} is spread approximately uniformly the microstate spaces. Besides this, even if we can choose an appropriate value of Y associated to each X{(n)}, it is unclear if Y{(n)} would be approximately uniformly distributed over its microstate space.

— Information geometry for types in the large-$n$ limit of random matrices (2501.00703 - Jekel, 1 Jan 2025) in Introduction, Motivation (after discussion of d_{W,CEP} and equation (1.1))