Fuglede--Kadison Determinants of Shifted Matrix Semicircles

Source abstract

We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle $S_a = a \otimes 1 + \sum_j A_j \otimes s_j$ with deterministic mean $a$. For every real $z$, $\log Δ(S_a - z) = -1/2 + \inf_b F_z(b)$, where the infimum is over positive definite matrices $b$ such that the map $X \mapsto bXb - \sum_j A_j X A_j$ is positive semidefinite, and $F_z$ is an explicit geodesically convex function. Both sides equal $-\infty$ exactly when $z$ is an atom of the spectral measure. The formula was proposed, without proof, in a recent post of Speicher. In the centered case $a=0$ it gives a new proof of the Mai-Speicher capacity formula, and by hermitization it yields a finite-dimensional variational formula for the logarithmic potential of the Brown measure of an arbitrary matrix-valued semicircular element. The proof identifies the determinant with the value of the Dyson action at its critical point and rewrites this critical-point problem as a saddle problem in three matrix variables. For a spectral parameter $z + i\varepsilon$ in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor $b_\varepsilon$ in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting $\varepsilon \to 0$ in the optimal values, with no compactness assumption on the minimizers. At points where the boundary value of the Cauchy transform is nondegenerate, the scales $b_\varepsilon$ converge and the infimum is attained at their limit; this minimizer lies on the boundary of the feasible set when $z$ is in the support of the spectral measure and in its interior when $z$ is in the resolvent set.