On The Eigenvalue Rigidity of the Laguerre Unitary Ensemble

Abstract: In this paper, we establish an optimal global rigidity estimate for the eigenvalues of the Laguerre unitary ensemble. Using the central limit theorem, we first construct a random measure via the eigenvalue counting function and then prove its convergence to a Gaussian multiplicative chaos measure, which yields the desired rigidity result. To prove this convergence, we apply a sufficient condition due to Claeys et al. [7] and carry out an asymptotic analysis of the corresponding exponential moments.

• The paper establishes optimal eigenvalue rigidity by proving two-sided bounds for eigenvalue fluctuations at the (log N)/N scale.
• It employs advanced probabilistic techniques, steep asymptotic analysis of Hankel determinants, and Riemann-Hilbert methods to control exponential moments.
• The findings reinforce the universality of extreme eigenvalue statistics in β=2 ensembles, extending applications to high-dimensional statistics and quantum systems.

Eigenvalue Rigidity in the Laguerre Unitary Ensemble

Introduction and Context

This work establishes optimal global eigenvalue rigidity estimates for the Laguerre Unitary Ensemble (LUE) with a general one-cut regular potential $V(x)$. The LUE serves as a canonical example of a random matrix ensemble with eigenvalue density:

$$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$

where $V$ is real-analytic, one-cut, and regular on $[-1, \infty)$. The investigation aligns the global scale eigenvalue deviations in LUE with those known for other classical ensembles such as the GUE, JUE, and CUE.

Eigenvalue rigidity concerns the fine-scale deviations of individual eigenvalues $\lambda_j$ from their "classical locations" $\kappa_j$ defined by percentiles of the limiting equilibrium measure $\mu_L$. Previous works, particularly by Erdős, Yau, and collaborators, provided upper tails for general Wigner ensembles but lacked optimal lower bounds. Optimality at the $(\log N)/N$ scale was exhibited first in the CUE using log-correlated field techniques and Gaussian multiplicative chaos (GMC), as established by Arguin, Belius, Bourgade and others. This paper fills the gap for LUE and, by extension, further solidifies the conjectured universality of optimal global rigidity for $\beta=2$ ensembles.

Methodology Overview

The core of the analysis utilizes advanced probabilistic techniques, steep asymptotic analysis of Hankel determinants, and Riemann-Hilbert problem (RHP) methods. The proof essentially proceeds via the following steps:

1. Log-correlated Field Structure and Central Limit Theorem: The authors consider the appropriately normalized eigenvalue counting function $h_N(x)$ and demonstrate that it converges in the sense of central limit theorems (CLT) to a log-correlated Gaussian field $$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$0 with covariance:

$$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$1

This exploits deep results on linear statistics in polynomial ensembles.

2. Gaussian Multiplicative Chaos and GMC-CLT Link: Building on the framework of Claeys, Fahs, Lambert, and Webb [CFL2021], the analysis constructs a random measure

$$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$2

and verifies its convergence, as $$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$3, to the GMC associated with the limiting log-correlated field, thereby connecting extreme value statistics of $$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$4 to GMC theory.

1. Control of Exponential Moments via Hankel Determinants: The moment generating functionals of $$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$5 are related by Heine's identity to Hankel determinants with Fisher-Hartwig singularities at the jump points. The authors analyze the large-$$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$6 asymptotics of these determinants in various regimes (bulk, edge, and merging singularities) by deriving and exploiting RHPs amenable to Deift-Zhou nonlinear steepest descent techniques. This step is intricate and requires refined matching of global and local parametrices near singular features in the spectrum.
2. Maximum Statistics and Rigidity Extraction: Using the connection between the maximum of $$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$7 and the GMC, both upper and lower tail probabilities for the maximal deviations of eigenvalues from their classical locations are extracted, with particular care given to behavior near the spectral edges, where fluctuations are more subtle due to vanishing density.
3. Iterative Edge Refinement: An iterative "refinement" argument is developed to handle the regime near the hard ($$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$8) and soft ($$\rho_N(\lambda_1, \ldots, \lambda_N) = \frac{1}{Z_N} \prod_{1\leq k<j \leq N} |\lambda_k - \lambda_j|^2 \prod_{j=1}^N (1+\lambda_j)^\alpha e^{-NV(\lambda_j)}, \quad \lambda_j > -1,$$9) spectral edges, where the naive Taylor expansion approximations break down due to singular behavior of the equilibrium density. This is coupled with explicit control on higher-order error terms in the bulk-to-edge transition.

Main Results

Optimal Global Rigidity

The principal finding is a two-sided optimal bound for eigenvalue fluctuations:

$V$0

for any $V$1, where $V$2 is the distribution function of the equilibrium measure $V$3, and $V$4 encodes local density variations, especially near the edges.

This not only matches the order but achieves matching coefficients up to $V$5 slack, establishing the result as optimal.

Extreme Behavior of the Counting Function

A parallel result holds for the extreme value of the cumulative deviation $V$6:

$V$7

which characterizes the maximum increment of the empirical distribution function over its limiting equilibrium counterpart.

These results fully extend the optimal fluctuation statements previously established for the GUE, JUE, and CUE [CFL2021, ABB2017, Dai-Lu-JUE].

Technical Innovations and Analytical Contributions

The analysis generalizes and synthesizes several deep techniques:

• Establishes the necessary asymptotics for Hankel determinants with Fisher-Hartwig singularities in very close proximity and near edge singularities, which requires merging local and global RHP analysis—particularly relevant for controlling exponential moments of $V$8 in non-bulk regimes.
• Adopts and extends the sufficient conditions from [CFL2021] for convergence of exponential moment measures to the GMC in the context of the LUE, which exhibits a qualitatively different density behavior at the hard edge.
• Introduces an iterative refinement methodology for the control of eigenvalue fluctuations in shrinking edge neighborhoods, exploiting continuity and jump structure in the counting function, and leveraging monotonicity and singular scaling of the equilibrium measure.

Implications and Future Directions

The proof that LUE possesses the same optimal global rigidity as the GUE, JUE, and CUE—at the $V$9 scale with matching leading constant—provides strong evidence toward the conjectured universality of extreme eigenvalue rigidity in $[-1, \infty)$0 one-cut regular ensembles. The techniques further suggest that the GMC-based approach is a robust analytical framework for extreme statistics in random matrix theory and related log-gas models.

Practically, these results refine probabilistic estimates necessary for understanding fine spectral features in high-dimensional statistics, wireless communications (random covariance models), and quantum information (entanglement spectra). Theoretically, the extension of this method to broader classes (multi-cut, non-unitary invariant ensembles, general $[-1, \infty)$1) and the refinement of matching coefficients in other regimes remain substantial open directions. Additionally, precise control of characteristic polynomial maxima, and the study of the full extremal process beyond the maximum, are natural sequelae leveraging the connections to log-correlated fields and Gaussian chaos.

Conclusion

This work rigorously establishes that the global eigenvalue fluctuations in the Laguerre Unitary Ensemble match the optimal $[-1, \infty)$2 order—both upper and lower—for all eigenvalues across the spectrum, up to the edges. The convergence of the associated random measure to GMC underpins the extreme behavior, and the detailed asymptotic analysis of the relevant Hankel determinants via RHP methods provides the key technical engine. The results substantially advance the universality program in eigenvalue rigidity and further cement the centrality of log-correlated random field and GMC paradigms in random matrix theory.

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References (for arXiv context):

• T. Claeys, B. Fahs, G. Lambert, C. Webb, "How much can the eigenvalues of a random Hermitian matrix fluctuate?", Duke Math. J. 170 (2021), no. 9, 2085–2235. (Gustafsson et al., 2020)
• L.P. Arguin, D. Belius, P. Bourgade, "Maximum of the characteristic polynomial of random unitary matrices", Comm. Math. Phys. 349 (2017), 703–751. (Arguin et al., 2015)
• D. Dai, C. Lu, "On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble", (Dai et al., 24 Nov 2025)
• C. Charlier, R. Gharakhloo, "Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities", Adv. Math. 383 (2021), 107672. (Vuijk et al., 2020)