1,721,091 research outputs found
The spectral dimension and geometrical universality on graphs
No full textThe spectral dimension d̄ of an infinite graph, defined according to the asymptotic behavior of the Laplacian operator spectral density, seems to be the right generalization of the Euclidean dimension d of lattices to non translationally invariant networks when dealing with dynamical and thermodynamical properties. In fact d̄ exactly replaces d in most laws where dimensional dependence explicitly appears: the spectrum of harmonic oscillations, the average autocorrelation function of random walks, the critical exponents of the spherical model, the low temperature specific heat, the generalized Mermin-Wagner theorem, the infrared singularities of the Gaussian model and many other. Still, d̄ would be a rather unsatisfactory generalization of d if it hadn't a second fundamental property: the independence of geometrical details at any finite scale (or geometrical universality). Here we show that d̄ is invariant under all geometrical transformations affecting only finite scale topology. In particular we prove that d̄ is left unchanged by any quasi-isometry (including coarse-graining and addition of finite range couplings), by local rescaling of couplings and by addition of infinite range of couplings provided they decay faster than a given power law
Fast Rare Events in Exit Times Distributions of Jump Processes
Get PDFRare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat-tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology, and finance: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on timescales orders of magnitude smaller than the typical timescale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations
Single-big-jump principle in physical modeling
No full textThe big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell
Rare events in extreme value statistics of jump processes with power tails
No full textWe study rare events in the extreme value statistics of stochastic symmetric jump processes with power tails in the distributions of the jumps, using the big -jump principle. The principle states that in the presence of stochastic processes with power tails statistics, if at a certain time a physical quantity takes on a value much larger than its typical value, this large fluctuation is realized through a single macroscopic jump that exceeds the typical scale of the process by several orders of magnitude. In particular, our estimation focuses on the asymptotic behavior of the tail of the probability distribution of maxima, a fundamental quantity in a wide class of stochastic models used in chemistry to estimate reaction thresholds, in climatology for earthquake risk assessment, in finance for portfolio management, and in ecology for the collective behavior of species. We determine the analytical form of the probability distribution of rare events in the extreme value statistics of three jump processes with power tails: L & eacute;vy flights, L & eacute;vy walks, and the L & eacute;vy-Lorentz gas. For the L & eacute;vy flights, we re-obtain through the big-jump approach recent analytical results, extending their validity. For the L & eacute;vy-Lorentz gas, we show that the topology of the disordered lattice along which the walker moves induces memory effects in its dynamics, which influences the extreme value statistics. Our results are confirmed by extensive numerical simulations
Rare events in generalized Lévy Walks and the Big Jump principle
No full textThe prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations and with complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction
Dynamical dimension splitting on fractals: Structures with different diffusive and vibrational spectral dimensions
No full textOn fractals and inhomogeneous structures that have been studied up to now, a single parameter, the spectral dimension, rules diffusion and vibrational dynamics. However, in principle, two distinct parameters could be necessary to describe the two physical phenomena. In this paper we show the existence of fractal structures where two different spectral dimensions are required. Random walks and vibrational spectrum are studied by renormalization group as well as alternative techniques to obtain the exact values of the dimensions and to clarify the origin of such dynamical dimension splitting. © World Scientific Publishing Company
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