The most isolated eigenvalue in a random-matrix model has a sharply describable largest gap, according to a new arXiv preprint. The theorem identifies the greatest distance from an eigenvalue in a fixed bulk region to its nearest other eigenvalue with a Gumbel law, a mathematical probability law for an extreme value.
The result concerns the complex Ginibre ensemble: matrices whose entries are independent, centered complex Gaussian variables with variance 1/n. It is a theorem-based asymptotic analysis of a random-matrix model rather than an analysis of observed data.
A formula for the rarest spacing
The analysis derives an explicit fourth-power centering and an equivalent affine normalization under which the largest gap converges to the same Gumbel law. It also provides the correction terms needed to locate the extreme gap on the scale predicted by the model.
The centering includes logarithmic, log-log and constant corrections. The reported numerical approximations for two constants are a1 = −2.3014159133… and a0 = 1.0063109967….
The result extends beyond the single largest gap. For any fixed rank among the extremes, the corresponding transformed order statistic has the stated Poisson-order-statistic limiting law.
Tracking an anchored empty neighbourhood
To study a large gap, the proof anchors the calculation at an eigenvalue and examines the chance that a surrounding neighbourhood contains no other points. Its central object is the reduced-Palm hole probability of the infinite Ginibre process: in plain terms, the chance of finding no other point around a chosen anchor.
The proof combines an exact tail coordinate with a sharp large-radius expansion of that hole probability, an adaptive-radius Poisson approximation, and a deterministic inversion that turns the hole exponent back into a location for the largest gap.
The hole expansion is carried through the constant term, with a remainder of O(r^{-1/6}) in the large-radius regime stated by the theorem. The analysis also compares finite- and infinite-matrix reduced-Palm holes with relative exponential accuracy throughout the bulk and across the radii relevant to the maximum.
A bulk result, not an edge prediction
The theorem applies to a fixed bulk set B with positive finite area, zero boundary measure and strict containment inside the unit disk. Eigenvalues in B are compared with their nearest neighbours from the full spectrum, rather than only with points that also lie inside B.
The conclusions are asymptotic in matrix size. The paper reports no finite-size uncertainty assessment, numerical simulation or empirical validation, so the explicit normalization is not presented as a finite-matrix guarantee.
The analysis is specific to the complex Gaussian Ginibre model and does not establish the same result at the spectral edge or for other random-matrix ensembles. Whether the explicit centering remains accurate at finite sizes is also not addressed.
A mathematical preprint
The document is an arXiv version 1 preprint dated 20 August 2026. Its stated contribution is a complete extreme-value description of the largest bulk gap, including its centering through constant order.
The author reports support from the Swedish Foundation for International Cooperation in Research and Higher Education under grant PD2023-9315.
Paper data and sources
Original title: Largest bulk gap of the complex Ginibre ensemble
Authors: Philippe Moreillon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text