@article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Poisson-process limit laws yield Gumbel max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.022129}, pages = {12}, year = {2019}, abstract = {"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j)entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the min-max. The limit laws hold whenever the entries' distribution has a density, and yield highly applicable approximation tools and design tools for the max-min and min-max of large random matrices. A brief of the results presented herein is given in: Gumbel central limit theorem for max-min and min-max}, language = {en} } @article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Gumbel central limit theorem for max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.020104}, pages = {6}, year = {2019}, abstract = {The max-min and min-max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the max-min and min-max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit laws—akin to the central limit theorem—for the max-min and min-max of large random matrices. The limit laws intertwine random-matrix theory and extreme-value theory, couple the matrix dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix entries' distribution. Due to their generality and universality, as well as their practicality, these results are expected to have a host of applications in the physical sciences and beyond.}, language = {en} } @article{KrapfMetzler2019, author = {Krapf, Diego and Metzler, Ralf}, title = {Strange interfacial molecular dynamics}, series = {Physics today}, volume = {72}, journal = {Physics today}, number = {9}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0031-9228}, doi = {10.1063/PT.3.4294}, pages = {48 -- 54}, year = {2019}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Transport in exclusion processes with one-step memory: density dependence and optimal acceleration}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {38}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab37e4}, pages = {19}, year = {2019}, abstract = {We study a lattice gas of persistent walkers, in which each site is occupied by at most one particle and the direction each particle attempts to move to depends on its last step. We analyse the mean squared displacement (MSD) of the particles as a function of the particle density and their persistence (the tendency to continue moving in the same direction). For positive persistence the MSD behaves as expected: it increases with the persistence and decreases with the density. However, for strong anti-persistence we find two different regimes, in which the dependence of the MSD on the density is non-monotonic. For very strong anti-persistence there is an optimal density at which the MSD reaches a maximum. In an intermediate regime, the MSD as a function of the density exhibits both a minimum and a maximum, a phenomenon which has not been observed before. We derive a mean-field theory which qualitatively explains this behaviour.}, language = {en} } @article{PalyulinBlackburnLomholtetal.2019, author = {Palyulin, Vladimir V. and Blackburn, George and Lomholt, Michael A. and Watkins, Nicholas W. and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.}, title = {First passage and first hitting times of Levy flights and Levy walks}, series = {New journal of physics : the open-access journal for physics}, volume = {21}, journal = {New journal of physics : the open-access journal for physics}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab41bb}, pages = {23}, year = {2019}, abstract = {For both L{\´e}vy flight and L{\´e}vy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For L{\´e}vy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the L{\´e}vy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Correlations and transport in exclusion processes with general finite memory}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab47fb}, pages = {31}, year = {2019}, language = {en} } @misc{PalyulinBlackburnLomholtetal.2019, author = {Palyulin, Vladimir V and Blackburn, George and Lomholt, Michael A and Watkins, Nicholas W and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.}, title = {First passage and first hitting times of L{\´e}vy flights and L{\´e}vy walks}, series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, number = {785}, issn = {1866-8372}, doi = {10.25932/publishup-43983}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-439832}, pages = {25}, year = {2019}, abstract = {For both L{\´e}vy flight and L{\´e}vy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For L{\´e}vy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the L{\´e}vy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.}, language = {en} } @article{AydinerCherstvyMetzler2019, author = {Aydiner, Ekrem and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Money distribution in agent-based models with position-exchange dynamics}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {92}, journal = {The European physical journal : B, Condensed matter and complex systems}, number = {5}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2019-90674-0}, pages = {4}, year = {2019}, abstract = {Wealth and income distributions are known to feature country-specific Pareto exponents for their long power-law tails. To propose a rationale for this, we introduce an agent-based dynamic model and use Monte Carlo simulations to unveil the wealth distributions in closed and open economical systems. The standard money-exchange scenario is supplemented with the position-exchange agent dynamics that vitally affects the Pareto law. Specifically, in closed systems with position-exchange dynamics the power law changes to an exponential shape, while for open systems with traps the Pareto law remains valid.}, language = {en} } @article{PadashChechkinDybiecetal.2019, author = {Padash, Amin and Chechkin, Aleksei V. and Dybiec, Bartlomiej and Pavlyukevich, Ilya and Shokri, Babak and Metzler, Ralf}, title = {First-passage properties of asymmetric Levy flights}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {45}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab493e}, pages = {48}, year = {2019}, abstract = {L{\´e}vy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of L{\´e}vy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering L{\´e}vy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of L{\´e}vy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.}, language = {en} } @article{VojtaSkinnerMetzler2019, author = {Vojta, Thomas and Skinner, Sarah and Metzler, Ralf}, title = {Probability density of the fractional Langevin equation with reflecting walls}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.042142}, pages = {11}, year = {2019}, abstract = {We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications.}, language = {en} }