177,205 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
No full textRare 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
A two-populations Ising model on diluted random graphs
No full textWe consider the Ising model for two interacting groups of spins embedded in an Erdös–Rényi random graph. The critical properties of the system are investigated by means of extensive Monte Carlo simulations. Our results evidence the existence of a phase transition at a value of the inter-groups interaction coupling J12C which depends algebraically on the dilution of the graph and on the relative width of the two populations, as explained by means of scaling arguments. We also measure the critical exponents, which are consistent with those of the Curie–Weiss model, hence suggesting a wide robustness of the universality class
Rohlin distance and the evolution of influenza A virus: weak attractors and precursors.
No full textThe evolution of the hemagglutinin amino acids sequences of Influenza A virus is studied by a method based on an informational metrics, originally introduced by Rohlin for partitions in abstract probability spaces. This metrics does not require any previous functional or syntactic knowledge about the sequences and it is sensitive to the correlated variations in the characters disposition. Its efficiency is improved by algorithmic tools, designed to enhance the detection of the novelty and to reduce the noise of useless mutations. We focus on the USA data from 1993/94 to 2010/2011 for A/H3N2 and on USA data from 2006/07 to 2010/2011 for A/H1N1. We show that the clusterization of the distance matrix gives strong evidence to a structure of domains in the sequence space, acting as weak attractors for the evolution, in very good agreement with the epidemiological history of the virus. The structure proves very robust with respect to the variations of the clusterization parameters, and extremely coherent when restricting the observation window. The results suggest an efficient strategy in the vaccine forecast, based on the presence of "precursors" (or "buds") populating the most recent attractor
Effective target arrangement in a deterministic scale-free graph
No full textWe study the random-walk problem on a deterministic scale-free network, in the presence of a set of static, identical targets; due to the strong inhomogeneity of the underlying structure the mean first-passage time (MFPT), meant as a measure of transport efficiency, is expected to depend sensitively on the position of targets. We consider several spatial arrangements for targets and we calculate, mainly rigorously, the related MFPT, where the average is taken over all possible starting points and over all possible paths. For all the cases studied, the MFPT asymptotically scales like
∼N^θ, being N the volume of the substrate and θ ranging from 1−log2/log3, for central target(s), to 1, for a single peripheral target
Non-equilibrium critical properties ofthe Ising model on product graphs
No full textWe study numerically the non-equilibrium critical properties of the Ising model defined on direct products of graphs, obtained from factor graphs without phase transition (Tc = 0). On this class of product graphs, the Ising model features a finite temperature phase transition, and we find a pattern of scaling behaviors analogous to the one known on regular lattices: observables take a scaling form in terms of a function L(t) of time, with the meaning of a growing length inside which a coherent fractal structure, the critical state, is progressively formed. Computing universal quantities, such as the critical exponents and the limiting fluctuation-dissipation ratio X∞, allows us to comment on the possibility to extend universality concepts to the critical behavior on inhomogeneous substrates
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
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