Anomalous scaling and phase transition in large deviations of dynamical observables of stationary Gaussian processes

TL;DR

The paper addresses large deviations of the dynamical observables $A_n=\int_{0}^{T} x^n(t)\,dt$ for stationary Gaussian processes, revealing an anomalous $T$-scaling $P(A_n,T)\sim \exp[-T^{\mu} f_n(\Delta A_n T^{-\nu})]$ with $\mu,\nu<1$ and a first-order dynamical phase transition in the rate function. It develops a unified framework that covers both short- and long-range correlations, with exponents $\mu$ and $\nu$ determined by $n$ and the covariance tail $\alpha$ (or Hurst $H$ in fOU), and shows that localized instanton-like optimal paths drive the DPT except in very long-range regimes where a smooth crossover occurs. The work derives explicit expressions for the instanton action $c_n$ and the Gaussian fluctuations coefficient $\beta_n$, and it validates the theory via Replica-Exchange Wang-Landau simulations across Gaussian and fractional OU processes, including $H=1/2$ (standard OU). The results map out phase diagrams in $(\alpha,n)$ and $(H,n)$, highlighting when the DPT persists and when it vanishes, with implications for understanding condensation-like transitions in nonequilibrium fluctuations. Overall, the study advances the understanding of anomalous large-deviation scaling in Gaussian processes and provides a robust numerical approach to probe extremely rare events.

Abstract

We study large deviations, over a long time window $T \to \infty$, of the dynamical observables $A_n = \int_{0}^{T} x^n(t) dt$, $n=3,4,\dots$, where $x(t)$ is a centered stationary Gaussian process in continuous time. We show that, for short-correlated processes the probability density of $A_n$ exhibits an anomalous scaling $P(A_n,T) \sim \exp[-T^μ f_n(ΔA_n T^{-ν})]$ at $T\to \infty$ while keeping $ΔA_n T^{-ν}$ constant. Here $ΔA_n$ is the deviation of $A_n$ from its ensemble average. The anomalous exponents $μ$ and $ν$ depend on $n$ and are smaller than $1$, whereas the rate function $f_n(z)$ exhibits a first-order dynamical phase transition (DPT) which resembles condensation transitions observed in many systems. The same type of anomaly and DPT, with the same $μ$ and $ν$, was previously uncovered for the Ornstein-Uhlenbeck process - the only stationary Gaussian process which is also Markovian. We also uncover an anomalous behavior and a similar DPT in the long-correlated Gaussian processes. However, the anomalous exponents $μ$ and $ν$ are determined in this case not only by $n$ but also by the power-law long-time decay $\sim |t|^{-α}$ of the covariance. The different anomalous scaling behavior is a consequence of a faster-than-linear scaling with $T$ of the variance of $A_n$. Finally, for sufficiently long-ranged correlations, $α<2/n$, the DPT disappears, giving way to a smooth crossover between the regions of typical, Gaussian fluctuations and large deviations. The basic mechanism behind the DPT is the existence of strongly localized optimal paths of the process conditioned on very large $A_n$ and coexistence between the localized and delocalized paths of the conditioned process. Our theoretical predictions are corroborated by replica-exchange Wang-Landau simulations where we could probe probability densities down to $10^{-200}$.

Figures (7)

• Figure 1: Phase diagram of the anomalous scaling behavior of the distribution $P(A_n,T)\simeq \exp\left[-T^\mu f_n\left(\Delta A_n T^{-\nu}\right)\right]$ for $n>2$ on the $(\alpha, n)$ plane. Different colors correspond to regions with a different behavior of the scaling exponents $\mu$ and $\nu$. The green region represents the effectively short-correlated regime $\alpha> \alpha_c$, where the rate function $f_n(y)$ exhibits a first-order DPT identified in Ref. Smith2022. The yellow region corresponds to the moderately correlated regime with $\alpha_*\le \alpha<\alpha_c$, which also displays a first-order DPT, but with a different behavior of $\mu$ and $\nu$. The blue region corresponds to the long-correlated regime $\alpha<\alpha_*$, where the rate function $f_n(y)$ is analytic. For illustrative purposes we have extended the phase diagram to non-integer $n$, but the results are applicable only for integer $n>2$, as indicated by the dashed lines.
• Figure 2: Phase diagram of the anomalous scaling behavior of the distribution $P(A_n,T)$ for $n>2$ on the $(H, n)$ plane. Different colors correspond to regions with a different behavior of the scaling exponents $\mu$ and $\nu$. The green region represents the short-correlated regime $H\le H_c$, where the rate function exhibits a first-order DPT. The yellow region corresponds to the moderately correlated regime with $H_*\ge H>H_c$, which also displays a first-order DPT, but with a different behavior of $\mu$ and $\nu$. The blue region corresponds to the long-correlated regime $H>H_*$, where the DPT disappears. For illustrative purposes, we have extended the phase diagram to non-integer $n$, but the results are applicable only for integer $n>2$, as indicated by the dashed lines.
• Figure 3: The emerging first-order DPT for the Gaussian covariance. Left: the large-$T$ behavior of the rescaled action, $-T^{-1/2}\ln P(A_3,T)$, as measured in the REWL simulations for a set of values of $T\gg 1$ marked by different colors. Right: the first derivative of the rescaled action with respect to $A_3$. The red dots mark the predicted critical point $y_c$ (left) and the finite discontinuity of the derivative (right).
• Figure 4: The large-$T$ behavior of the rescaled action, $-T^{-\mu}\ln P(A_3,T)$, as measured in the REWL simulations of the fOU process for several values of $T$ (marked by different colors) and a set of Hurst exponents $H$: a) $H=1/4$; b) $H = 1/2$; c) $H=3/5$, and d) $H=3/4$. The red dots mark the predicted critical point $y_c$.
• Figure 5: The first derivative of the rescaled action, measured in the REWL simulations, with respect to $A_3$ for different Hurst exponents $H$: a) $H=1/4$; b) $H = 1/2$; c) $H=3/5$, and d) $H=3/4$. The red dots in the panels a-c mark the finite discontinuity of the derivative.