Eigenspectrum Decay (α–ReQ) in Complex Systems

• Eigenspectrum Decay (α–ReQ) is the study of how eigenvalues diminish with varying coupling strength, signaling critical transitions across mathematical and physical systems.
• The framework unifies analyses from nonequilibrium Markov processes, quantum Hamiltonians, and spectral graph theory, highlighting distinct regimes and universal phase transitions.
• In machine learning, controlling eigenspectrum decay in attention matrices enhances training stability and expressivity by regulating signal propagation and regularization.

Eigenspectrum decay, often denoted in the literature as “α–ReQ” problems or approaches, refers to the paper of how the eigenvalues (the spectrum) of operators—such as Markov generators, Schrödinger-type Hamiltonians, Dirac operators, or structured matrices in networks—decay or distribute under the influence of parameters that either regulate the strength or spatial decay of couplings. The parameter α typically indexes the decay, regularization, or coupling strength, and its variation is directly linked to critical transitions in system behavior, ranging from spectral phase transitions to optimal stabilization in dynamical systems.

1. Eigenspectrum Decay in Nonequilibrium Markov Generators

In nonequilibrium Markovian dynamics, the generator $L$ is generally non-self-adjoint, requiring a biorthogonal system of left- and right-eigenmodes $\{\psi_k\}$, $\{\phi_k\}$ satisfying $L\phi_k = -\lambda_k \phi_k$, $\psi_k L = -\lambda_k \psi_k$, and $\langle\psi_k, \phi_\ell\rangle = \delta_{k\ell}$. This generalization is crucial when $L$ does not enjoy detailed balance, as is typical in equilibrium cases. The orthonormality (or rather, biorthonormality) is defined with respect to a metric that incorporates the invariant measure $\pi$.

A central insight is that the Fisher-Rao metric on the space of statistical states arises naturally in this context. It provides a differential-geometric measure of distances on the space of parameterized probability distributions:

$$g_{ij} = \sum_x \pi(x;\theta) (\partial_{\theta_i}\log \pi(x;\theta))(\partial_{\theta_j}\log \pi(x;\theta)).$$

Degeneracy in the Fisher-Rao metric, i.e., $\det(g_{ij})=0$, signals that fluctuations along particular parameter directions become unobservable, indicating a critical transition. The eigenspectrum is then classified both by its real versus complex nature and by degeneracy, with the latter functioning as an indicator of phase transitions between nonequilibrium regimes (Polettini, 2011).

At criticality, order parameters $m(t)$—such as observable correlation functions—exhibit power-law decay $m(t)\sim t^{-\alpha}$, differentiating critical behavior from generic exponential decay. Near critical points, the variance of unbiased estimators diverges due to the vanishing Fisher information, $$\operatorname{Var}(\hat\theta_i) \gtrsim (g^{-1})_{ii}$$, indicating heightened sensitivity and breakdown of the standard statistical inference paradigm.

2. Eigenspectrum Decay in Quantum Hamiltonians with Decaying Potentials

Eigenspectrum decay is particularly explicit in self-adjoint operators subject to spatially decaying random or deterministic potentials. For a prototypical class, consider the one-dimensional Schrödinger operator $H = -d^2/dx^2 + a(x) F(X_x)$, where $a(x)\sim x^{-\alpha}$ governs the decay rate $\alpha>0$.

The scaling limit of the unfolded spectrum (eigenvalue spacings) exhibits three universal regimes (Kotani et al., 2012, Nakano, 2022, Bourget et al., 2020):

• Supercritical regime, $\alpha > 1/2$: The decaying potential is negligible at infinity. The spectrum is absolutely continuous, the local eigenvalue statistics converge to a clock process (regular, equally spaced behavior), and eigenfunctions are extended (Lebesgue limiting spatial measures).
• Critical regime, $\alpha = 1/2$: The transition point. Level statistics converge to the circular $\beta$-ensemble; eigenfunctions decay in a power-law fashion modulated by Brownian (log-correlated) fluctuations, corresponding to a random fractal spatial profile.
• Subcritical regime, $\alpha < 1/2$: The potential dominates at large $x$. The spectrum is almost surely pure point, exhibiting Poisson (fully localized) statistics, with eigenfunctions localized around random sites (delta measure limiting spatial profiles).

These transitions have direct analogues in discrete models (e.g., for Dirac operators (Bourget et al., 2020)) and in the behavior of random matrices and disordered systems, illustrating a robust universality across operator classes.

3. Analytical Structure and Regularization in Singular/Critical Cases

In singular cases, such as the Schrödinger operator with an inverse-square potential $q_\alpha(r) = (\alpha-1/4)/r^2$, eigenspectrum decay is best understood through analytic continuation in both the coupling constant $\alpha$ and boundary parameters. For $\alpha < 0$, there are infinitely many negative eigenvalues accumulating at $-\infty$ or $0$, forming—in the logarithmic energy variable—a nearly arithmetic sequence with spacing $2\pi/\sqrt{|\alpha|}$ (Smirnov, 2020). As $\alpha \to 0^-$, these eigenvalues coalesce with the continuum, and for $\alpha \geq 0$ only at most one discrete state remains. The spectral measure and generalized eigenfunctions depend analytically on $\alpha$, providing a framework that regularizes the classical singularity at $\alpha=0$ and captures the “phase transition” as infinitely many negative eigenvalues are absorbed into the continuous spectrum.

Similarly, time-frequency limiting operators and damped wave equations with singular damping possess explicit asymptotic spectral formulas that describe super-exponential or finite spectral decay, respectively. For example, the spectrum of the damped wave operator with singular damping $u_{tt} + (2\alpha/x)u_t = u_{xx}$ has finitely many negative eigenvalues when $\alpha \in \mathbb{N}$ and otherwise exhibits a complex tail, with the spectral abscissa determining the optimal energy decay rate (Freitas et al., 2020, Ammari et al., 28 Apr 2024, Bonami et al., 2015).

4. Matrix Eigenspectrum Decay: Graphs and Regularization

The α–ReQ paradigm extends to structured matrices in combinatorics and network theory. A one-parameter family $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ interpolates between the adjacency matrix and the signless Laplacian (Rojo, 2017). For specific graphs such as “bugs,” exact eigenspectra can be computed: the spectrum contains a “bulk” eigenvalue of multiplicity $(n-d-1)$ and the remaining eigenvalues (including the spectral radius) correspond to a symmetric tridiagonal matrix whose order increases with the diameter. As $\alpha$ varies, the decay, spread, and extremal behavior of the eigenvalues can be exactly characterized, with the spectral radius being maximal for bug graphs among all graphs of fixed order and diameter, for all $\alpha\in[0,1)$.

In quantum graph models, the spectrum’s discrete part controls exponential energy decay, with the spectral abscissa determining the system’s optimal stabilization rate (Ammari et al., 28 Apr 2024).

5. Eigenspectrum Concentration and Learning Dynamics in Machine Learning

Eigenspectrum decay also has foundational importance in data-driven and machine learning settings, notably in attention-based neural networks. The distribution and variance of the eigenvalues of the query-key (“QK”) parameter matrix $W$ control attention localization and gradient propagation (Bao et al., 3 Feb 2024). A small variance in the eigenspectrum, with non-vanishing mean, ensures localization of the signal propagation probability $\rho(\theta)$, optimizing both the expressivity and trainability of transformer models by simultaneously suppressing both rank collapse and entropy collapse failure modes. Formally,

$$d^2V[w_i] = d \operatorname{tr}(W^2) - |\operatorname{tr}(W)|^2,$$

and localization occurs when this variance is minimized relative to the mean. Regularization methods that explicitly penalize the spectral scale (e.g., “LocAteR”) enhance training stability and model performance.

6. Asymptotics and Counting Laws in Coupled and Decaying Spectra

In semiclassical and perturbative regimes, the growth rate in the number of discrete eigenvalues $N(\lambda,\alpha)$ drawn into the spectral gap by a decaying or growing coupling $\alpha$ is governed by the spatial decay properties of the perturbing potential $V(x)$ (Gao et al., 28 Jun 2025). For integrable $V$, $N(\lambda,\alpha)\sim (\alpha/4\pi)\int V(x)\,dx$; for potentials with $V(x)\sim \Psi(\theta)/|x|^p$, $$N(\lambda,\alpha)\sim (\alpha^{2/p}/4\pi)\int [(\lambda+\Psi(\theta)|x|^{-p})_+^2 - m^2]_+^{1/2}dx$$ as $\alpha\to\infty$. These asymptotic laws provide a precise quantitative measure of eigenspectrum decay (number and distribution of eigenvalues) as a function of coupling, connecting Weyl-type analysis with physical impurity models in materials such as graphene.

7. Summary and Broader Context

Eigenspectrum decay (α–ReQ) forms a unifying conceptual and technical paradigm across statistical mechanics, spectral theory, combinatorics, and machine learning. It captures:

• The geometrization of nonequilibrium criticality via the Fisher-Rao metric and statistical equivalence principles,
• Universal spectral phase transitions in disordered and decaying potential models,
• Rigorous asymptotic laws for the counting and distribution of spectral levels under spatially decaying perturbations,
• Concrete mechanisms for regularization and stability in both quantum systems (via analytic parametrization) and in the optimization of neural architectures.

The decay parameter α thus acquires a dual role: as a probe of statistical geometry (e.g., phase transitions, critical exponents) and as a “spectral regularizer” controlling the distribution, decay, and behavior of the spectrum in broad classes of mathematical and applied systems.