Eigenvalue Motion under Ginibre Perturbations

Definitions and mathematical statement

Let $N\in{\mathbb{N}}$ be positive, and let $M_N,M_N'\in{\mathbb{C}}^{N\times N}$ be independent complex Ginibre matrices with the same entry normalization. Here a complex Ginibre matrix has independent, identically distributed centered circular complex Gaussian entries. Writing their unspecified variance as $v_N>0$, each entry has independent real and imaginary parts with distribution $\mathcal N(0,v_N/2)$. The source does not specify $v_N$, including its possible dependence on $N$.

For a real parameter $\theta$, define \[ M_N(\theta)=\cos\theta\,M_N+\sin\theta\,M_N'. \] Write its eigenvalues, counted with algebraic multiplicity, as the unordered multiset \[ \Lambda_N(\theta)=\{\lambda_1(\theta),\ldots,\lambda_N(\theta)\}, \qquad \det(zI_N-M_N(\theta))=\prod_{i=1}^N(z-\lambda_i(\theta)). \] The two random matrices are sampled once and held fixed as $\theta$ varies.

Problem. Describe how the eigenvalues of $M_N(\theta)$ vary with $\theta$, that is, describe the random spectral motion $\theta\mapsto\Lambda_N(\theta)$. The source writes labeled trajectories $\lambda_i(\theta)$ but does not prescribe a labeling convention; individual branches may be considered locally on intervals of simple spectrum. The formula is defined for every $\theta\in{\mathbb{R}}$, although the source specifies no parameter interval or boundary conditions.

Unresolved scope. This is an exploratory question rather than a specified conjecture. The source does not select an exact finite-dimensional law, a particular trajectory statistic, or a large-$N$ limit with specified parameter and eigenvalue scaling. These choices, together with the entry normalization and any global labeling convention, require clarification before a more specific mathematical assertion can be stated.

Short English statement

Sample two independent complex Ginibre matrices and interpolate between them using cosine and sine coefficients. Describe the resulting motion of their eigenvalues as the real parameter varies. The source leaves the normalization, trajectory labeling, and precise probabilistic or asymptotic target unspecified.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003728$, title Eigenvalue Motion under Ginibre Perturbations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Strongly Correlated Random Interacting Processes (2019), Oberwolfach Reports, pp. 187--253, DOI 10.4171/owr/2018/4. Supplied source URL: https://doi.org/10.4171/owr/2018/4.
• Literature source labeled "original OWR report": https://doi.org/10.4171/OWR/2018/4.
• Literature source labeled "Chalker--Mehlig, eigenvector statistics": https://doi.org/10.1103/PhysRevLett.81.3367.
• Literature source labeled "Dysonian dynamics of the Ginibre ensemble": https://arxiv.org/abs/1403.7738.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem partially solved and reports that squared eigenvalue-velocity statistics and large-matrix dynamical descriptions are known, while a complete law or asymptotic description of the trajectories remains open. This is supplied dated curation derived from the OWR collection, not an independently verified assessment; the time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note reports manual semantic, typographic, and LaTeX review for self-containment and preservation of the target. This records the supplied review claim and does not establish mathematical correctness or independently verify the literature.