Analyticity in 1/N of expected traces for polynomial potentials

Determine whether, for multimatrix models Y^N with polynomial potential V, the expected normalized trace E[tr_N(P(Y^N))] is an analytic function of 1/N for every noncommutative polynomial P. Establishing such analyticity would justify applicability of the polynomial method in this setting.

Background

The authors survey several methods for proving strong convergence. The polynomial method yields sharp results when expected traces of polynomials are rational (or analytic) functions of N{-1}. They point out that in their general Ck setting such analyticity cannot be guaranteed and emphasize that even for polynomial potentials it is not clear whether this property holds.

Resolving this would clarify the scope of the polynomial method for multimatrix models arising from polynomial potentials and could connect asymptotic expansions with exact rational/analytic dependence on N{-1}.

References

The polynomial method relies on $\mathbb{E} \tr_N(P(YN))$ being an analytic function in $1/N$, which we will certainly not be able to guarantee when $V$ is merely a $\cCk$ function, and is even unclear if $V$ is polynomial, as was the case in .

— Asymptotic expansion for transport maps between laws of multimatrix models  (2604.03213 - Jekel et al., 3 Apr 2026) in Introduction (Section 1), discussion of the polynomial method