At lower energies, the resonances in scattering experiments are often isolated. In quantum chaotic many-body, disordered, or generically stochastic systems, the resonances overlap at larger energies. Eventually, the Ericson regime is reached in which the cross section behaves like a random function. The scattering-matrix elements then follow a universal Gaussian distribution. For more than 60 years, the emergence of this robust additional universal behavior on top of the universal system stochasticity has awaited a concise analytical treatment. We derive the transition to the Ericson regime in the universal Heidelberg approach and prove the universal Gaussian distribution by a proper asymptotic expansion. We also obtain explicit formulas for the moments and for the cross-section distribution. We compare with microwave experiments and numerical simulations.

Extreme values and the tail behavior of probability distributions are essential for quantifying and mitigating risk in complex systems of all kinds. In multivariate settings, accounting for correlations is crucial. Although extreme value analysis for infinite correlated systems remains an open challenge, we propose a practical framework for handling a large but finite number of correlated time series. We develop our approach for finance as a concrete example but emphasize its generality. We study the extremal behavior of high-frequency stock returns after rotating them into the eigenbasis of the correlation matrix. This separates and extracts various collective effects, including information on the correlated market as a whole and on correlated sectoral behavior from idiosyncratic features, while allowing us to use univariate tools of extreme value analysis. This holds even for high-frequency data where discretization effects normally complicate analysis. We employ a peaks-over-threshold approach and thereby fully avoid the analysis of block maxima. We estimate the tail shape of the rotated returns while explicitly accounting for nonstationarity, a key feature in finance and many other complex systems. Our framework facilitates tail risk estimation relative to larger trends and intraday seasonalities at both market and sectoral levels.

To understand the emergence of Ultrafast Extreme Events (UEEs), the influence of algorithmic trading or high-frequency traders is of major interest as they make it extremely difficult to intervene and to stabilize financial markets. In an empirical analysis, we compare various characteristics of UEEs over different years for the US stock market to assess the possible non-stationarity of the effects. We show that liquidity plays a dominant role in the emergence of UEEs and find a general pattern in their dynamics. We also empirically investigate the after-effects in view of the recovery rate. We find common patterns for different years. We explain changes in the recovery rate by varying market sentiments for the different years. Overall, our results hint at a certain degree of universal behavior.

Many-body Floquet models with exact space–time symmetry, such as the kicked Ising spin chain (KIC), provide natural examples of systems with dual-unitary dynamics. The requirement of exact space–time symmetry is, however, highly restrictive, as it permits only nearest-neighbor interactions. Based on a pair of Hadamard matrices, we construct a wide family of dual-unitary kicked spin chains with long-range interactions. We show that local two-point correlations in such models propagate along the light-cone edges |n| = r|t|, where r is the interaction range, and can be derived analytically for operators with local support. This approach is illustrated using the example of a kicked Ising spin chain with next-to-next-neighbor interactions.

Multivariate distributions are needed to capture the correlation structure of complex systems. In previous works, we developed a random matrix model for correlated multivariate joint probability density functions that accounts for the non-stationarity typically found in complex systems. Here, we apply these results to the returns measured in correlated stock markets. Only the knowledge of the multivariate return distributions allows for a full-fledged risk assessment. We analyze intraday data of 479 US stocks included in the S&P 500 index during the trading year of 2014. We focus particularly on the tails which are algebraic and heavy. The non-stationary fluctuations of the correlations make the tails heavier. With the few-parameter formulae of our random matrix model, we can describe and quantify how the empirical distributions change for varying time resolution and in the presence of non-stationarity.

In the last few years, the dynamical characterization of the power output of a wind turbine by means of a Langevin equation has been well established. For this approach, temporally highly resolved measurements of wind speed and power output are used to obtain the drift and diffusion coefficients of the energy conversion process. These coefficients fully determine a Langevin stochastic differential equation with Gaussian white noise. The drift term specifies the deterministic behavior of the system whereas the diffusion term describes the stochastic behavior of the system. A precise estimation of these coefficients is essential to understand the dynamics of the power conversion process of the wind turbine. We show that the dynamics of the power output of a wind turbine have a hidden dependency on turbine's different operational states. Here, we use an approach based on clustering Pearson correlation matrices for different observables on a moving time window to identify different operational states. We have identified five operational states in total, for example the state of rated power. Those different operational states distinguish non-stationary behavior in the mutual dependencies and represent different turbine control settings. As a next step, we condition our Langevin analysis on these different states to reveal distinctly different behaviors of the power conversion process for each operational state. Moreover, in our new representation hysteresis effects which have typically appeared in the Langevin dynamics of wind turbines seem to be resolved. We assign these typically observed hysteresis effects clearly to the change of the wind energy system between our estimated different operational states. In this contribution, we discuss further consequences for the meaning of hysteric switching and detection of malbehaviors in wind turbines.

We investigate the interaction of many wind turbines in a wind farm with a focus on their electrical power production. The operational data of two offshore wind farms with a ten minute and a ten second time resolution, respectively, are analyzed. For the correlations of the active power between turbines over the entire wind farms, we find a dominant collective behavior. We manage to subtract the collective behavior and find a significant dependence of the correlation structure on the spatial structure of the wind farms. We further show a connection between the observed correlation structures and the prevailing wind direction. We attribute the differences between the two wind farms to the differences in the turbine spacing within the two wind farms.

At lower energies, the resonances in scattering experiments are often isolated. The crucial parameter is the ratio of average resonance width and average mean level spacing. Towards larger energies, this parameter grows, because the resonances overlap. Eventually the cross-section becomes a random function and the scattering matrix elements follow a universal Gaussian distribution. For more than sixty years, this Ericson transition awaits a concise analytical treatment. We provide a complete solution within the Heidelberg approach which provides a full-fledged model of the scattering process. As a side result, we obtain explicit formulae for the moments of the distributions. We compare with microwave experiments.

We report on experimental studies of the distribution of the off-diagonal elements of the scattering (S) matrix of open microwave networks with symplectic symmetry and chaotic wave dynamics. These consist of two geometrically identical subgraphs with unitary symmetry described by complex conjugate Hamiltonians that are coupled by a pair of bonds. The results are compared to random-matrix theory (RMT) predictions obtained on the basis of the Heidelberg approach for the S matrix of open quantum-chaotic systems, employing random matrices from the Gaussian symplectic ensemble. We demonstrate that deviations observed in the distributions of the off-diagonal S-matrix elements may be attributed to the fact that the subgraphs are not fully connected, and propose a RMT model, which takes this into account and indeed confirms the experimental results.

Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant variables, especially in the presence of non-stationarity. We use a generalized scalar product between correlation matrices to clearly demonstrate this non-stationarity. Further, we present a model that we recently put forward, which captures how the non-stationary fluctuations of correlations make the tails of multivariate distributions heavier. Here, we provide the resulting formulae including Gaussian or Algebraic features. Compared to our previous results, we manage to remove in the Algebraic cases one out of the two, respectively three, fit parameters which considerably facilitates applications. We demonstrate the usefulness of these results by deriving joint distributions for linear combinations of amplitudes and validating them with financial data. Furthermore, we explicitly work out the moments of our model distributions. In a forthcoming paper we apply the model to financial markets.

Why does a traffic jams form out of nowhere, and why does it stretch for kilometers even after the initial cause is passed? This study examines how congestion moves and spreads across motorways using a surprisingly simple method: response functions. These functions are based purely on data and show how changes in traffic flow, density, and velocity are connected over time and space. Using real-world data from German motorways, we track how traffic reacts to earlier disturbances, capturing the waves of slowing and accelerating that drivers experience in stop-and-go traffic. The results demonstrate how congestion propagates and how its rhythm can be measured and predicted. Unlike complex traffic models, this approach requires no simulations or assumptions about driver behavior. It works directly from the information provided by the road. The goal is clear: to understand congestion better so that we can manage it more effectively and perhaps spend less time stuck in it.

The winding number is the topological invariant that classifies chiral symmetric Hamiltonians with one-dimensional parametric dependence. In this work we complete our study of the winding number statistics in a random matrix model belonging to the chiral unitary class AIII. We show that in the limit of large matrix dimensions the winding number distribution becomes Gaussian. Our results include expressions for the statistical moments of the winding number and for the k-point correlation function of the winding number density.

Traders on financial markets generate non-Markovian effects in various ways, particularly through their competition with one another which can be interpreted as a game between different (types of) traders. To quantify the market mechanisms, we empirically analyze self-response functions for pairs of different stocks and the corresponding trade sign correlators. While the non-Markovian dynamics in the self-responses is liquidity-driven, it is expectation-driven in the cross-responses which is related to the emergence of correlations. We empirically study the non-stationarity of these responses over time. In our previous data analysis, we only investigated the crisis year 2008. We now considerably extend this by also analyzing the years 2007, 2014 and 2021. To improve statistics, we also work out averaged response functions for the different years. We find significant variations over time revealing changes in the traders' game.

We consider the problem of local correlations in the kicked, dual-unitary coupled maps on D-dimensional lattices. We demonstrate that for D>=2, fully dual-unitary systems exhibit ultra-local correlations: the correlations between any pair of operators with local support vanish in a finite number of time steps. In addition, for $D=2$, we consider the partially dual-unitary regime of the model, where the dual-unitarity applies to only one of the two spatial directions. For this case, we show that correlations generically decay exponentially and provide an explicit formula for the correlation function between the operators supported on two and four neighbouring sites.

The causal connection between congestions and velocity changes at different locations induces various statistical features, which we identify and measure in detail. We carry out an empirical analysis of large-scale traffic data on a local motorway network around the Breitscheid intersection in the North Rhine-Westphalia, Germany. We put forward a response function which measures the velocity change at a certain location versus time conditioned on a congestion at another location. We use a novel definition of the corresponding congestion indicator to ensure causality. We find that the response of velocities to the congestion exhibits phase changes in time. A negative response at smaller time lags transforms into positive one at larger time lags, implying a certain traffic mechanism. The response decays as a power law with the distance. We also identify a scaling property leading to a collapse of the response functions on one curve.

When travelling by car from one location to another, our route is constrained by the road network. The network distance between the two locations is generally longer than the geodetic distance as the crow flies. We report a systematic relation between the statistical properties of these two distances. Empirically, we find a robust scaling between network and geodetic distance distributions for a variety of large motorway networks. A simple consequence is that we typically have to drive 1.3 ± 0.1 times longer than the crow flies. This scaling is not present in standard random networks; rather, it requires non-random adjacency. We develop a set of rules to build a realistic motorway network, also consistent with the above scaling. We hypothesise that the scaling reflects a compromise between two societal needs: high efficiency and accessibility on the one hand, and limitation of costs and other burdens on the other.

We further pursue a recently tested approach to the study of transport systems. Using empirical data collected on German motorways, we investigate the use of response functions as a tool to study traffic dynamics in motorway networks. We use a response function that measures the change in traffic observed at a given location over time conditioned on congestion at another location. By introducing several velocity-dependent congestion indicators, we are able to work out the basic characteristics of the flow and density responses to congestion. Our results demonstrate that these responses capture the propagation of congestion, the acceleration phases of traffic participants during congestion, and the correlation between the three fundamental traffic observables: flow, density, and velocity. Furthermore, a direct comparison with the underlying data reveals the potential for characterizing traffic patterns with response functions. A relationship between the shape of the response functions and the traffic oscillation during a traffic jam can be demonstrated. Moreover, our findings allow us to determine the propagation velocity of congestion from responses. Importantly, in the course of our analysis, we also identify a scaling property that results in the collapse of the response functions onto a single curve.

The understanding of congestions contributes to the development of effective traffic management strategies. The propagation of congestions from a motorway section to the neighboring ones results in correlations. Here, we study symmetrized time-lagged correlation matrices and show how their spectral properties reveal congestion durations. We first carry out an empirical analysis of velocities for two local motorway networks, then set up a numerical simulation for indicator time series of traffic phases, and further propose a simplified, analytical model capturing various scenarios of congestion durations. Our empirical analysis reveals a transition behavior for the dominant eigenvalue as function of time lags, reflecting changes in traffic dynamics. Furthermore, both the numerical simulations and the analytical model disclose a nonlinear relation between the spectral transition and the congestion duration.

We consider an Ising model with quenched surface disorder, the disorder average of the free energy is the main object of interest. Explicit expressions for the free energy distribution are difficult to obtain if the quenched surface spins take values of +/- 1. Thus, we choose a different approach and model the surface disorder by Gaussian random matrices. The distribution of the free energy is calculated. We chose skew-circulant random matrices and analytically compute the characteristic function of the free energy distribution. From the characteristic function we numerically calculate the distribution and show that it becomes log-normal for sufficiently large dimensions of the disorder matrices, and in the limit of infinitely large matrices tends to a Gaussian. Furthermore, we establish a connection to the central limit theorem.