Linearized Correlation Functions

• Linearized correlation functions are computed by expanding fields around a background solution, capturing small fluctuations to reveal key response characteristics and universal properties.
• They simplify complex nonlinear dynamics into manageable linear equations, yielding explicit analytical expressions that clarify symmetries and conservation laws.
• They are widely applied across quantum field theory, holography, random matrix theory, and statistical physics, providing practical tools for response estimation and algorithmic simulation.

Linearized correlation functions are correlation functions computed or analyzed by expanding the system of interest around a background (often a classical solution or fixed point) and then retaining terms up to linear order in the fluctuations. In quantum field theory, statistical mechanics, hydrodynamics, holography, random matrix theory, and dynamical systems, studying linearized correlation functions is a fundamental step for extracting information such as response functions, universality properties, spectral statistics, and symmetries. The linearization process simplifies the often intractable nonlinear problem to a manageable form, frequently yielding explicit analytical expressions, mapping to determinantal or universal structures, and revealing salient features of the system under paper.

1. Definition and Scope of Linearized Correlation Functions

Linearized correlation functions refer to correlation functions evaluated in the regime where the underlying degrees of freedom are expanded to linear order around a background solution, equilibrium, or classical ground state. This procedure typically involves:

• Expanding fields, operators, or observables as $\Phi = \Phi_0 + \delta\Phi$, where $\Phi_0$ is the background and $\delta\Phi$ is the small fluctuation.
• Retaining terms up to quadratic order in the action, Hamiltonian, or equations of motion, so that the equations governing $\delta\Phi$ are linear.
• Constructing correlation functions (e.g., two-point, multi-point) of the form $\langle \delta\Phi(x) \delta\Phi(y) \rangle$, which can often be computed exactly due to Gaussianity.

Linearized correlation functions capture response properties, reveal the structure of symmetries and conservation laws (such as Ward identities), and in many cases govern the leading behavior of physical observables in perturbative and some nonperturbative regimes.

2. Holography and Linearized Correlation Functions in Non-AdS Backgrounds

In the holographic context, particularly for non-asymptotically AdS backgrounds such as Schrödinger spacetimes, the linearized analysis is essential for establishing the holographic dictionary, controlling renormalization, and extracting dual operator correlators. Key features include:

• Identifying dual sources via the radial expansion: For Schrödinger backgrounds, sources correspond to specific powers of the radial coordinate $e^{-{\Delta_-} r}$ up to any finite order in the deformation parameter $b$ (Rees, 2012).
• Holographic renormalization at linearized level: Divergences in the gravitational on-shell action are canceled by adding counterterms constructed from the leading terms in the linearized expansion. These counterterms include both single-trace (local) and multi-trace (nonlocal or scheme-dependent) structures.
• Computation of linearized two-point functions: Regular solutions to the linearized bulk equations (typically Bessel function solutions) are matched to boundary data to yield explicit momentum- or position-space correlators; e.g., for operators dual to a massive vector field, $$\langle \mathcal{O}_u(u,v) \mathcal{O}_u(0) \rangle \sim 1/(u^4 v^2)$$.
• Scheme dependence: The necessity of multi-trace counterterms introduces finite, scheme-dependent parameters in the final answer.
• Modified Ward identities: Linearization around nontrivial backgrounds or with deformations (e.g., null vector $b$ deformation) leads to modified conservation laws and trace relations.

This methodology ensures controlled, explicit extraction of field theory correlators even in settings with irrelevant operators (non-renormalizable deformations) and non-relativistic symmetries.

3. Linearization in Quantum Field Theory and Many-Body Systems

Linearized correlation functions are extensively used to analyze quantum fields on curved backgrounds and in many-body systems:

• In de Sitter backgrounds for linearized quantum gravity (Morrison, 2013), the linearized two-point graviton correlator is explicitly constructed in both transverse-traceless and generalized de Donder gauges. The construction ensures manifest de Sitter invariance, adherence to the Hadamard condition (correct short-distance singularity), and provides the necessary Green's functions for quantum corrections and loop calculations.
• In classical and quantum statistical systems, linearizing the hydrodynamic or kinetic equations yields propagative, diffusive, or critical modes whose correlation functions determine temporal and spatial responses. For example, in the Lebwohl-Lasher model (Varghese et al., 2017), linearizing the continuity equations for the nematic director and angular momentum yields coupled, propagative fluctuation modes in the nematic phase and purely diffusive ones in the isotropic phase, with implications for long-time tail behavior of autocorrelation functions.

These approaches facilitate analytic computation, uncover universality (e.g., exponential decay governed by the cosmic no-hair theorem), and clarify the impact of symmetries and conservation laws at linear order.

4. Linearized Correlation Functions in Random Matrix Theory and Spectral Statistics

Random matrix theory provides a rich context in which linearization and universal correlation functions appear:

• For Gaussian random normal matrices (Riser, 2013), the joint eigenvalue correlation functions are determined by the asymptotics of reproducing kernels constructed from orthogonal polynomials. The “linearization” involves rescaling coordinates around bulk points at the scale of typical eigenvalue separations ($\sim 1/\sqrt{n}$), yielding universal determinantal formulas in the limit $n\to\infty$:

$$K_\text{univ}(a, b) = \frac{1}{\pi} \exp\left(-\frac{1}{2}|a-b|^2 + i \operatorname{Im}(\bar{a}b) \right).$$

• At the edge of the spectrum, similar scaling properties yield a transition from constant density to a complementary error function profile.
• In non-Hermitian and correlated matrix ensembles (Jana et al., 28 Mar 2025), linear eigenvalue statistics (LES) and their fluctuations are controlled through linearization (cumulant expansions, resolvent bounds), with universality in the limiting distributions captured through systematic control of derivative terms.

These universal results underlie the statistical character of spectral fluctuations in complex systems and relate directly to linearized properties of the underlying stochastic processes.

5. Bayesian, Probabilistic, and Algorithmic Approaches to Linearized Correlation Estimation

Modern data-driven fields leverage linearization for optimal correlation estimation and efficient simulation:

• Bayesian estimation frameworks for correlation functions (Gutierrez-Rubio et al., 2022) exploit the fact that for stationary Gaussian processes, Fourier space “linearizes” the estimation of spectral parameters: each mode decouples, making posterior distributions tractable and providing full uncertainty quantification.
• Probabilistic models in blind source separation and independent component analysis (ICA) (Sasaki et al., 2015) incorporate linearly structured dependencies and allow for efficient, quadratic-score matching to recover dependency structures.
• Algorithmic generation of correlated noise with prescribed spectra is facilitated by the linearity of the underlying Gaussian and Fourier-transformed models.

These approaches have practical utility in time series analysis, signal processing, and high-dimensional data analysis, where the linearized framework yields both statistically optimal estimators and natural generative algorithms.

6. Combinatorial and Algebraic Linearization in Integrable Systems and Field Theory

• In integrable systems such as the XX0 Heisenberg spin chain (Bogoliubov et al., 2021), linearization manifests combinatorially through determinantal representations in terms of Schur polynomials and symmetric function theory. Correlation functions are “linearized” by reducing to determinants, whose entries reflect the linear action of algebraic structures.
• In recent developments in quantum A-infinity algebraic formulations for correlation functions (Konosu et al., 17 May 2024), the linearization occurs at the level of algebraic operators—correlation functions are computed as coefficients of linear operators (matrix elements) arising from the full, nonperturbative operator inverse, bypassing the usual split into free and interacting parts. This enables both perturbative and genuinely nonperturbative computation, with explicit control over “linearized” contributions corresponding to distinct Lefschetz thimble sectors.

Such algebraic and combinatorial techniques allow comprehensive, efficient calculation and connect the linearization in analysis to that in algebraic and enumerative settings.

7. Impact, Universality, and Future Directions

Linearized correlation functions serve as the principal analytical tools for extracting universal features of complex systems, elucidating the effect of symmetries, and constructing the leading description in a wide variety of fields, including holography, quantum chaos, signal processing, and random matrix theory. They provide the basis for both rigorous proofs of universality and practical algorithms for estimation and simulation. Extensions to nonlinear, nonperturbative, and strongly correlated regimes frequently leverage the linearized results as benchmarks or starting points for more sophisticated treatments.

Future work includes developing higher-order (beyond-linear) analyses, robust nonperturbative frameworks (e.g., quantum homotopy algebras), and systematic calibration of linearized predictions within fully nonlinear, data-driven, or algorithmic contexts. The persistent appearance of determinantal, universal, and algebraic structures in linearized correlation functions underscores their foundational role across mathematical physics, probability, and applied mathematics.