The Liouville and the intersection properties are equivalent for planar graphs

It is shown that if a planar graph admits no non-constant bounded harmonic functions then the trajectories of two independent simple random walks intersect almost surely.Comment: 4 pages, 1 figur

Monopole clusters at short and large distances

We present measurements of various geometrical characteristics of monopole clusters in SU(2) lattice gauge theory. The maximal Abelian projection is employed and both infinite, or percolating cluster and finite clusters are considered. In particular, we observe scaling for average length of segments of the percolating cluster between self-crossings, correlators of vacuum monopole currents, angular correlation between links along trajectories. Short clusters are random walks and their spectrum in length corresponds to free particles. At the hadronic scale, on the other hand, the monopole trajectories are no longer random walks. Moreover, we argue that the data on the density of finite clusters suggest that there are long-range correlations between finite clusters which can be understood as association of the clusters with two-dimensional surfaces, whose area scales.Comment: 9 pages, 11 figure

Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories

Continuous time random walks have been developed as a straightforward generalisation of classical random walk processes. Some 10 years ago, Fogedby introduced a continuous representation of these processes by means of a set of Langevin equations [H. C. Fogedby, Phys. Rev. E 50 (1994)]. The present work is devoted to a detailed discussion of Fogedby's model and presents its application for the robust numerical generation of sample paths of continuous time random walk processes.Comment: 7 pages, 7 figure

Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks

A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are also provided

L\'evy flights versus L\'evy walks in bounded domains

L\'evy flights and L\'evy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of L\'evy flights is associated with their pathological physical properties, which in turn are resolved by the concept of L\'evy walks. Here, we explore L\'evy flights and L\'evy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.Comment: 15 pages, 13 figure

Anomalous Lineshapes and Aging Effects in Two-Dimensional Correlation Spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times