Singularities of matrix semicircles

Abstract

Let $S=\sum_{i=1}^r A_i\otimes s_i$ be a matrix semicircular element, with Hermitian coefficients $A_i\in M_n(\mathbb{C})$ and free standard semicircular generators $s_i$. Its scalar spectral density $f$ is governed, through Speicher's equation (a matrix Dyson equation), by the completely positive covariance map $\eta_S(X)=\sum_i A_iXA_i$. We treat the singular regime, where the pencil $\sum_i A_i x_i$ is full but not semisimple and $f$ is unbounded at the origin, in contrast to the bounded real-analytic density of the regular case. We prove three results. (i) The leading singularity exponent at $0$ is invariant under congruence $A_i\mapsto bA_ib^{*}$ of the pencil ($b$ invertible), and more generally under symmetric scaling of the covariance map. (ii) For binary elements ($r=2$) we obtain a complete classification: in Lancaster-Rodman canonical form every indecomposable cell is of one of three types, and $f(x)\sim c|x|^{-(n^*-1)/(n^*+1)}$ as $x\to0$ with an explicit constant $c$, where the exponent depends only on the size of the largest Jordan block (the effective chain length $n^*$) and not on the coupling. With (i) and the direct-sum behaviour, this classifies all full binary Hermitian pencils. (iii) The spectral classification is strictly coarser than the algebraic one: a Type III cell of size $2m$ with non-real $\beta$ and the direct sum of two Type II cells of size $m$ with parameter $|\beta|$ have identical scalar densities, yet their covariance maps are not symmetrically scalable; the scalar spectrum cannot detect the phase of $\beta$. Each type calls for a different method: a reduction of Speicher's equation to an autonomous discrete Painlev\'e I (McMillan) map (Type I), a Lyapunov-Schmidt reduction at the branch point (Type II), and a gauge reduction by a diagonal unitary (Type III).