arXiv subjects

Marc Höll

Publications and source records attributed to Marc Höll.

5 recordsLinked to original sources

arXiv2023

Controls that expedite first passage times in disordered systems

First passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first passage time of biased processes in different models, and employ the big jump principle that shows the dominance of the maximum trapping time on the first passage time. Inspired by the restart paradigm, we demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.

cond-mat.stat-mech↗

arXiv2021

Big jump principle for heavy-tailed random walks with correlated increments

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into physical sciences where systems usually exhibit correlations. Here, we study the principle for a random walk with correlated increments. Examples are the autoregressive model of first order and the discretized Ornstein-Uhlenbeck process both with heavy-tailed noise. The correlation leads to the dependence of large values of the sum not only on the big jump but also on the following increments. We describe this behaviour by two big jump principles, namely unconditioned and conditioned on the step number when the big jump occurs. The unconditional big jump principle is described by a correlation dependent shift between the sum and maximum distribution tails. For the conditional big jump principle, the shift depends also on the step number of the big jump.

cond-mat.stat-mech↗

arXiv2020

Large deviations of the ballistic Lévy walk model

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a `light' cone $-v_0 t<x<v_0 t$. In particular we study this density close to its maximum in the vicinity of the `light' cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations far from the `light' cone. However this law blows up in the vicinity of the `light' cone horizon which is nonphysical, in the sense that any finite time observation will never diverge. We claim that one can find two laws for the spatial density, the first one is the mentioned Lamperti-arcsine law describing the central part of the distribution and the second is an infinite density illustrating the dynamics for large $x$. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short `time' using a rate formalism.

cond-mat.stat-mech↗

arXiv2020

Extreme value theory for constrained physical systems

We investigate extreme value theory for physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a non-analytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in generality, i.e. even far from the thermodynamic limit. For example for renewal processes, the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example for the renewal process, we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalisable scaling function for the typical statistics and a non-normalised state describing the rare events.

cond-mat.stat-mech↗

arXiv2018

Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average

We present a general framework of detrending methods of fluctuation analysis of which detrended fluctuation analysis (DFA) is one prominent example. Another more recently introduced method is detrending moving average (DMA). Both methods are constructed differently but are similarly able to detect long-range correlations as well as anomalous diffusion even in the presence of nonstationarities. In this article we describe their similarities in a general framework of detrending methods. We establish this framework independently of the definition of DFA or DMA but by investigating the failure of standard statistical tools applied on nonstationary time series, let these be intrinsic nonstationarities such as for Brownian pathes, or external ones due to additive trends. In particular, we investigate the sample averaged mean squared displacement of the summed time series. By modifying this estimator we introduce a general form of the so-called fluctuation function and can formulate the framework of detrending methods. A detrending method provides an estimator of the fluctuation function which obeys the following principles: The first relates the scaling behaviour of the fluctuation function to the stochastic properties of the time series. The second principles claims unbiasedness of the estimatior. This is the centerpiece of the detrending procedure and ensures that the detrending method can be applied to nonstationary time series, e.g. FBM or additive trends. Both principles are formulated and investigated in detail for DFA and DMA by using the relationship between the fluctuation function and the autocovariance function of the underlying stochastic process of the time series.

cond-mat.stat-mech↗