Abrupt Decorrelation in Stochastic Systems

• Abrupt decorrelation is the phenomenon in stochastic systems defined by a sharp loss of statistical dependence as a control parameter crosses a critical threshold.
• It is mathematically characterized using statistical distances (e.g., KL divergence, Wasserstein) that show a sudden transition from maximal correlation to independence.
• Applications span models like LSDEs, random matrices, KPZ growth, and deep learning, offering insights into phase transitions and memory loss.

Abrupt decorrelation is a phenomenon in stochastic systems and high-dimensional random models characterized by an exceptionally sharp and well-defined loss of statistical dependence between coordinates, events, or observables as a control parameter (time, separation, system size, etc.) is varied. Unlike gradual exponential or algebraic decay, abrupt decorrelation is distinguished by a sudden drop from strong correlation to near-independence over a critical scale—an effect that has deep connections to the cut-off phenomenon in Markov chains, phase transitions, and universal scaling limits. Across stochastic processes, random matrix theory, statistical physics, and applied domains such as deep learning, abrupt decorrelation provides a unified framework for understanding how and when systems "forget" their initial state or become statistically independent.

1. Mathematical Characterization of Abrupt Decorrelation

The central mathematical approach is to compare, for a family of stochastic processes indexed by a parameter (often noise strength or system size), the joint law of variables at two locations in time or space—$\mathcal{L}_t^{(\epsilon)}$ for $\left(X_0^{(\epsilon)},X_t^{(\epsilon)}\right)$—against the law when the variables are independent, $\mathcal{I}_t^{(\epsilon)}$. A statistical distance $d^{(\epsilon)}(t)$ (for example, Kullback–Leibler, Wasserstein, or total variation) quantifies the dependence. Abrupt decorrelation is defined by

$$\lim_{\epsilon\to0}d^{(\epsilon)}(c\,t_\epsilon) = \begin{cases} M, & 0 < c < 1, \ 0, & c > 1, \end{cases}$$

for some critical time scale $t_\epsilon$ and maximal possible value $M$.

Additionally, one studies the profile

$$G(r):=\lim_{\epsilon\to0} d^{(\epsilon)}(t_\epsilon + r\,w_\epsilon)$$

for a window scale $w_\epsilon = o(t_\epsilon)$, revealing an explicit transition from maximal to zero dependence as $r$ sweeps from $-\infty$ to $+\infty$.

A prototypical example is the one-dimensional Ornstein–Uhlenbeck process

$$dX_t^{(\epsilon)} = -\theta X_t^{(\epsilon)}\,dt + \epsilon\,dB_t,$$

where $t_\epsilon \sim |\ln\epsilon|/\theta$ is the abrupt decorrelation time for the dependence between $X_0^{(\epsilon)}$ and $X_t^{(\epsilon)}$, as formally established by explicit computation of both KL-divergence and Wasserstein-2 distance. In multivariate systems, this transition is governed by the real part and Jordan block structure of the drift matrix (López et al., 20 Sep 2025).

2. Statistical Mechanics and Disordered Systems: Anderson Models and Random Matrices

In the context of the discrete Anderson model and its generalizations with finite-rank perturbations, abrupt decorrelation is realized spatially: eigenvalues associated with two distinct energies $E$ and $E'$ in the localization regime become independent as the system size grows. Quantitative estimates show that the probability of finding eigenvalues simultaneously near $E$ and $E'$ decays as $C(\ell/L)^{2d} (\log L)^C$, and the local statistics at these energies become asymptotically independent Poisson (or compound Poisson for finite-rank perturbations) processes (Klopp, 2010, Hislop et al., 2015). The mechanism is that localized eigenstates couple to disjoint regions of disorder, and so their dependence vanishes sharply as the system volume increases or energy difference exceeds a critical threshold. This is in contrast with extended systems (e.g., in Wigner matrices) where long-range correlations are universal.

For Wigner matrices and their minors, as the difference $k$ between matrix size $N$ and its minor $n = N - k$ surpasses $k \sim N^{2/3}$, there is an abrupt decorrelation transition: the largest eigenvalues (and corresponding eigenvectors) of the two matrices shift from being tightly correlated (for $k \ll N^{2/3}$) to statistically independent (for $k \gg N^{2/3}$). In the critical regime, the joint law of rescaled eigenvalues is given by a universal determinantal formula involving Airy functions (Bao et al., 9 Mar 2025). This establishes a precise threshold for abrupt decorrelation as a function of rank perturbation.

3. Nonlinear Growth Models and KPZ Universality

In growth models and particle systems within the Kardar–Parisi–Zhang (KPZ) universality class, decorrelation may be abrupt in space or time, though in these models the "abruptness" can also take the form of "slow decorrelation": under proper KPZ scaling, the rescaled interface fluctuations remain statistically identical along characteristics over time intervals of length $t^\nu$ with $\nu<1$, even as $t\to\infty$. The spatial process at time $tT$ and at $tT + t^{\nu}$ are asymptotically identical in law when observed at KPZ scaling. This "frozen" fluctuation structure is equivalent to a temporal abruptness, as the fluctuations suddenly decorrelate only when time increments exceed $t$ itself (Corwin et al., 2010).

Spatially, for the KPZ equation with narrow wedge data, the spatial covariance of the height field decays only as $t/x$ for large separation $x$, a "slow" but non-integrable decay. Despite the weakly decaying tail, for averaged observables (e.g., spatial averages over $[0,N]$), abrupt decorrelation manifests as a Gaussian (Brownian) CLT for properly rescaled sums—with a normalization by $(N\log N)^{1/2}$ reflecting the nonlocal nature of fluctuations (Gu et al., 29 Jun 2025).

4. Markov Chains, Percolation, and Memory Loss

Abrupt decorrelation is closely linked to the cut-off phenomenon in Markov processes: a Markov chain exhibits a sharp, drop-off in total variation distance from stationarity at a well-defined cut-off time. Similarly, for Markov processes and reversible chains, the instantaneous pairwise decorrelation of an event $C$—measured by $$\mathbb{P}[\omega_0, \omega_t\in C] - \mathbb{P}[\omega_0\in C]^2$$—implies a superpolynomial or exponential decay in the probability that $C$ occurs at all times in $[0,t]$ (Hammond et al., 2011). This "exit time" perspective formalizes abrupt loss of memory: the system remains correlated only up to a critical time, after which the occurrence of $C$ at all times becomes extremely unlikely.

5. Applications across Physical and Data-Driven Domains

The concept of abrupt decorrelation is seen in a range of physical and engineering settings:

• Wave Scattering in Complex Media: In optical and acoustic multiple scattering, measured correlations of the backscattered intensity field change abruptly as the sample thickness or scatterer size increases. Surface (thin-layer) and bulk (diffusive) contributions can be quantitatively separated, and their balance determines the sharpness of the decorrelation transition. Experimental autocorrelation measurements reveal a rapid crossover in behavior that is accurately captured by a superposition model, with further confirmation from fluctuation (noise) statistics that display a quasi-power law scaling (Zhang et al., 15 Apr 2025).
• Climate Proxy Records: In paleoclimate time series derived from Greenland ice cores, abrupt climatic shifts (such as Dansgaard–Oeschger events) are linked not only to changes in the deterministic drift function (bistability for atmospheric proxies; monostability for temperature proxies) but also to noise statistics. For the temperature proxy, pronounced higher-order Kramers–Moyal coefficients (e.g., $D_4(x)$) imply discontinuous, non-Gaussian forcing, producing sudden, "abrupt" transitions between states despite an underlying monostable drift (Riechers et al., 12 Feb 2025).
• Deep Learning and Reinforcement Learning: Abrupt decorrelation is relevant when enforcing exact independence among features (e.g., via whitening or SVD-based constraints in deep CCA). These hard constraints are computationally expensive and may introduce undesirable optimization dynamics. Recent approaches use "soft" decorrelation—such as stochastic decorrelation losses—that gently penalize correlation, thereby preventing abrupt, brittle loss of dependence while maintaining training efficiency (Chang et al., 2017, Mavrin et al., 2019). In deep RL, adding decorrelation terms to the loss can yield a rapid ("abrupt") improvement in learning as the latent space representation quickly transitions from highly redundant to informative.

6. Implications, Theoretical Insights, and Connections

Abrupt decorrelation brings several conceptual and technical insights:

• It provides a precise language and toolkit for quantifying "loss of memory" or "loss of dependence" not just through second moment calculations but via strong statistical distances and entire law comparisons.
• It establishes clean thresholds—sometimes sharp enough to define a universal transition—for decorrelation in diverse models, unifying the understanding of memory loss across time, space, scale, and type of observable.
• The phenomenon is often driven by the interplay between deterministic drift (e.g., exponential decay of fundamental solutions or drift matrices) and noise statistics (e.g., stochastic forcing, discontinuous jumps).
• The critical scales for abrupt decorrelation (e.g., $|\ln\epsilon|/\theta$ for LSDEs, $k\sim N^{2/3}$ for Wigner minors, $t^\nu$ for KPZ scaling) are determined by the underlying decay rates and system geometry.

The link to cut-off theory in Markov chains is not superficial: abrupt decorrelation in continuous-time stochastic dynamics is driven by exponential decay mechanisms whose transition from "correlated" to "fully mixed" occurs on an $O(|\ln \epsilon|)$ scale, paralleling the cut-off times seen in finite Markov chains.

7. Limitations and Future Directions

While abrupt decorrelation has been rigorously established in LSDEs, localized random Schrödinger operators, random matrices, and some statistical mechanics models, several open questions and directions remain:

• Extending these results to nonlinear, non-Gaussian, or non-Markovian systems.
• Sharpening the understanding of window profiles and universality classes of decorrelation transitions, particularly in non-equilibrium or high-dimensional settings.
• Investigating the operational consequences in high-dimensional data modeling (e.g., for explainability in neural networks or robustness in RL).
• Exploring connections to phase transitions and critical phenomena, especially in disordered systems where disorder-induced localization and delocalization transitions coincide with abrupt changes in correlation structure.

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Table 1: Prototypical Contexts for Abrupt Decorrelation

Context	Critical Scale or Parameter	Nature of Transition
1D/Multivariate LSDEs	$t_\epsilon \sim |\ln\epsilon|/\theta$	Temporal: sharp in $t$
Anderson models / Schrödinger operators	$|E - E'| >$ threshold	Spatial/energy: independent
Wigner minor process	$k \sim N^{2/3}$	Size: eigenvalues/eigenvectors decouple
KPZ growth processes	Time increment $t^\nu$, $\nu<1$	Space-time: frozen fluctuations
Markov processes / dynamical percolation	Continuous time $t$	Temporal: exit time drops
Complex wave scattering	Thickness $z_0$, particle size	Structural: from surface to volumetric decorrelation

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Abrupt decorrelation thus encapsulates a spectrum of sharp, rigorously quantifiable transitions from dependence to independence in stochastic and high-dimensional systems, with deep theoretical and practical ramifications in probability, spectral theory, statistical mechanics, and data science.