Ensemble NXS: AGN Variability & RMT Insights

• Ensemble NXS is a statistical metric that quantifies intrinsic variability by averaging normalized excess variances across a population.
• It is computed from single-source variance estimates, corrected for noise and time-scale effects, to probe AGN properties and black-hole scaling relations.
• Advanced methods like Bayesian hierarchical modeling are used to address bias, propagate errors, and manage selection effects in diverse datasets.

Ensemble normalised excess variance (commonly abbreviated as ensemble NXS, eNEV, or $\langle\sigma^2_\mathrm{NXS}\rangle$) is a central statistical observable in both time-domain astrophysics and random matrix theory, quantifying the intrinsic fractional variability or number variance across a population or ensemble. In extragalactic astrophysics, it is a key metric for probing the variability of active galactic nuclei (AGN) and constraining black-hole scaling relations. In random matrix theory, it quantifies particle number fluctuations—normalized by the mean—within spectral point processes such as the Ginibre ensembles. This article surveys the rigorous definition, estimation, and physical applications of ensemble normalised excess variance across these domains, with a focus on its use in AGN X-ray variability studies, statistical methodology, and universal properties in random matrix ensembles.

1. Formal Definition and Single-Object Excess Variance

For a discrete time series or light curve comprised of $N$ measurements $X_i$ (e.g., fluxes, count rates), with measurement errors $\sigma_i$ and mean $\mu$, the normalised excess variance is defined as

$$\sigma^2_\mathrm{NXS} = \frac{1}{N\mu^2} \sum_{i=1}^N \left[ (X_i - \mu)^2 - \sigma_i^2 \right]$$

This statistic estimates the intrinsic—i.e., Poisson (or photon-counting noise) subtracted—fractional variance for a single source over the sampled interval (Georgakakis et al., 2021, Vagnetti et al., 2016, Bogensberger et al., 2024, Georgakakis et al., 9 Dec 2025). In the absence of measurement noise, it reduces to a normalized sample variance. The subtraction of the estimated measurement variances $\sigma_i^2$ ensures that only intrinsic source fluctuations are measured, removing bias from stochastic noise.

2. Transition from Single-Object to Ensemble Excess Variance

For a population of $M$ sources (e.g., AGN in a flux-limited sample), each with its own normalised excess variance estimate $\sigma^2_{\mathrm{NXS},j}$, the ensemble (population-averaged) normalised excess variance is computed as

$$\langle \sigma^2_\mathrm{NXS} \rangle = \frac{1}{M} \sum_{j=1}^M \sigma^2_{\mathrm{NXS},j}$$

In AGN variability studies, one often considers ensemble averages in bins of X-ray luminosity $L_X$, black-hole mass $M_\mathrm{BH}$, or Eddington ratio $\lambda_\mathrm{Edd}$, sometimes weighted by selection effects or detection probabilities (Georgakakis et al., 2021, Georgakakis et al., 9 Dec 2025).

In populations with heteroscedastic variance or upper limits, a hierarchical model is applied: the distribution of single-source variances is modeled (typically as a log-normal), and the ensemble NXS is defined as the median or mean of this distribution, with scatter and uncertainty rigorously propagated (Georgakakis et al., 9 Dec 2025, Bogensberger et al., 2024). For random matrix point processes, the analogue is $\chi(R) = \mathrm{Var} N(R) / \langle N(R) \rangle$, where $N(R)$ counts eigenvalues within a prescribed region.

3. Methodological Issues: Corrections, Bayesian Estimation, and Biases

Bias Correction and Rest-Frame Duration

The value of $\sigma^2_\mathrm{NXS}$ depends sensitively on the timescale sampled, especially for processes with correlated (“red-noise”) variability. In AGN, $\sigma^2_\mathrm{NXS}$ grows as a power law with observer-frame or rest-frame duration $\Delta t$:

$$\langle \sigma^2_\mathrm{NXS} \rangle \propto \Delta t_\mathrm{rest}^a \qquad a \approx 0.12-0.20$$

Thus, different time samplings or different redshifts must be corrected by rescaling individual $\sigma^2_\mathrm{NXS}$ values to a common reference timescale. The standard correction is (Vagnetti et al., 2016):

$$\hat{\sigma}^2_{\mathrm{NXS}} = \sigma^2_{\mathrm{NXS}} \left( \frac{\widehat{\Delta t}}{\Delta t_\mathrm{rest}} \right)^{2b}$$

where $b$ is inferred from the ensemble structure function, typically $b \approx 0.12$.

Bayesian and Hierarchical Estimation

For sparse, Poisson-limited data (e.g., eROSITA epochal surveys), classical estimators of excess variance may be negative or underdispersed. Bayesian excess variance estimation (“bexvar”) improves robustness: photon-count data are modeled as arising from an underlying log-normal process, with full marginalization over Poisson likelihoods and population parameters. For an ensemble, this is extended hierarchically, with the population variance distribution hyperparameters inferred directly from the data (Bogensberger et al., 2024, Georgakakis et al., 9 Dec 2025).

Error Propagation and Scatter

The ensemble error incorporates both measurement uncertainties and intrinsic stochastic scatter. In log space,

$$\sigma_\mathrm{ens}^2 = \frac{1}{M^2}\sum_{j=1}^M \delta_j^2 + \frac{\Delta_s^2(N_b, \langle \sigma^2 \rangle)}{M}$$

where $\Delta_s$ characterizes finite-segment scatter, calibrated on stochastic simulations (Bogensberger et al., 2024). Survival analysis or upper limit treatment is required for censored samples.

4. Applications in Astrophysics: Probing Black-Hole Scaling and Accretion Physics

Ensemble NXS is a core observable for quantifying X-ray variability in AGN populations. Its dependence on X-ray luminosity, black-hole mass, and Eddington ratio encodes accretion physics via the PSD, as well as demographic scaling relations.

Connection to PSD Models

AGN X-ray variability follows a PSD described by a bending power law:

$\mathrm{PSD}(\nu) = A \nu^{-1}(1+\nu/\nu_b)^{-1}$

with break frequency $\nu_b$ and normalization $A$. Empirical results demonstrate that a PSD normalization $A \propto \lambda_\mathrm{Edd}^{-0.8}$ is required to reproduce observed ensemble NXS–luminosity trends (Georgakakis et al., 2021).

Constraining Black-Hole–Galaxy Scaling

Comparisons of $\langle \sigma^2_\mathrm{NXS} \rangle(L_X)$ to model predictions simultaneously constrain the black-hole mass–stellar mass relation. Models employing the dynamical scaling relation of Savorgnan & Graham (2016) are statistically favored by the ensemble NXS data (Georgakakis et al., 2021).

Empirical Trends

Key findings include:

• $\langle \sigma^2_\mathrm{NXS} \rangle$ declines with increasing $L_X$, anti-correlates with $M_\mathrm{BH}$, and, at fixed $M_\mathrm{BH}$, displays a non-monotonic dependence on $\lambda_\mathrm{Edd}$—declining up to $\lambda_\mathrm{Edd} \sim 0.1$, then rising again at high Eddington ratios, likely due to new variability mechanisms in highly accreting supermassive black holes (Georgakakis et al., 9 Dec 2025).
• The functional form is empirically fit as a 2D polynomial in $\log M_\mathrm{BH}$ and $\log \lambda_\mathrm{Edd}$.

These results robustly link ensemble NXS to the demographics and physics of accreting black holes.

5. Statistical Properties and Universality in Random Matrix Theory

In the random matrix context, ensemble normalised excess variance is defined for eigenvalue point processes as

$$\chi(R) = \frac{\mathrm{Var} N(R)}{\langle N(R) \rangle}$$

where $N(R)$ is the number of eigenvalues inside a disk of radius $R$. Exact results for complex, real, and symplectic Ginibre ensembles yield:

• In the origin (small-$R$) limit, $\chi(R) \to 1$ as expected for Poissonian statistics.
• In the bulk ($R \to \infty$), $\chi(R) \sim c(\beta)/R$ (with $c(\beta)$ ensemble-dependent), reflecting hyperuniform number variance scaling (Akemann et al., 2023).
• The same normalized variance structure recurs in bulk and edge scaling limits, reflecting statistical universality across Ginibre ensembles and physical realizations in cold atom experiments.

6. Practical Workflows and Survey Implementation

Robust measurement of ensemble NXS is operationally central in modern X-ray survey pipelines. The recommended methodology is as follows (Vagnetti et al., 2016, Bogensberger et al., 2024, Georgakakis et al., 9 Dec 2025):

1. Compute per-source $\sigma^2_\mathrm{NXS}$ (corrected for bias and rest-frame duration as appropriate, e.g., Eq. 8–9 in (Vagnetti et al., 2016)).
2. For sparse/low-count sources, use Bayesian or hierarchical estimators (e.g., “bexvar” or eBExVar).
3. Propagate both measurement and intrinsic scatter errors, especially for ensemble means.
4. Apply selection-corrected ensemble averaging, e.g., geometric mean in log space, with proper treatment of limits and non-Gaussian uncertainty.
5. In AGN population studies, map ensemble NXS as a function of $L_X$, $M_\mathrm{BH}$, and $\lambda_\mathrm{Edd}$, and fit to empirical or theoretical models to extract scaling relations.

A table summarizing relationships:

Ensemble NXS Domain	Definition / Observable	Population Trend
AGN Variability (astrophys.)	$\langle \sigma^2_\mathrm{NXS} \rangle$	$\downarrow$ with $L_X$, $M_\mathrm{BH}$
Ginibre RMT (spectral stat.)	$\chi(R) = \mathrm{Var} N(R)/\langle N(R) \rangle$	$\sim 1/R$ in the bulk

7. Implications, Limitations, and Future Prospects

Ensemble normalised excess variance plays a crucial role in confronting and constraining models of accretion variability, scaling relations, and universality classes of fluctuation statistics. Contemporary X-ray surveys have revealed new non-monotonic trends at high Eddington ratios, suggesting a role for super-Eddington flows and dynamic coronae not captured by standard PSD-based variability models (Georgakakis et al., 9 Dec 2025).

Limitations include residual dependencies on red-noise process timescales, observer cadence, and sensitivity to selection effects. Bayesian hierarchical and structure-function-based corrections are necessary for unbiased estimation, especially for heterogeneous or incomplete datasets.

As ongoing and future time-domain surveys (e.g., eROSITA, Athena) deliver larger and more diverse variability samples, ensemble NXS will remain central to the quantitative interface between population demographics, accretion physics, and universal fluctuation statistics (Bogensberger et al., 2024, Georgakakis et al., 2021, Georgakakis et al., 9 Dec 2025, Vagnetti et al., 2016, Akemann et al., 2023).