Fuglede--Kadison Determinants of Shifted Matrix Semicircles
Abstract
We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle Sa = a ⊗ 1 + ∑j Aj ⊗ sj with deterministic mean a. For every real z, log Δ(Sa − z) = −1/2 + infb Fz(b), where the infimum is over positive definite matrices b such that the map X ↦ bXb − ∑j Aj X Aj is positive semidefinite, and Fz is an explicit geodesically convex function. Both sides equal −∞ 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 + iε in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor bε in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting ε → 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 bε 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.
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BibTeX
@misc{kargin2026fugledekadison,
title = {{Fuglede--Kadison Determinants of Shifted Matrix Semicircles}},
author = {Vladislav Kargin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15180},
eprint = {2610.15180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fuglede--Kadison Determinants of Shifted Matrix Semicircles
AU - Vladislav Kargin
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15180
UR - https://arxiv.org/abs/2610.15180
AB - We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle S_a = a ⊗ 1 + ∑_j A_j ⊗ 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 ↦ bXb − ∑_j A_j X A_j is positive semidefinite, and F_z is an explicit geodesically convex function. Both sides equal −∞ 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 + iε in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor b_ε in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting ε → 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_ε 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.
ER -