In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the ...promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue
http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on
containing articles which show incredible possibilities of the CTRWs.
We establish cutoff for a natural random walk ( RW ) on the set of perfect matchings ( PM s), based on "rewiring". An n- PM is a pairing of 2 n objects. The k- PM RW selects k pairs uniformly at ...random, disassociates the corresponding 2 k objects, then chooses a new pairing on these 2 k objects uniformly at random. The equilibrium distribution is uniform over all n- PM s. The 2- PM RW was first introduced by Diaconis and Holmes (Proc. Natl. Acad. Sci. USA 95 (1998) 14600–14602; Electron. J. Probab. 7 (2002) no. 6), seen as a RW on phylogenetic trees. They established cutoff in this case. We establish cutoff for the k- PM RW whenever 2 ≤ k ≪ n . If k ≫ 1 , then the mixing time is n k log n to leading order. Diaconis and Holmes (Electron. J. Probab. 7 (2002) no. 6) relate the 2- PM RW to the random transpositions card shuffle. Ceccherini-Silberstein, Scarabotti and Tolli (J. Math. Sci. 141 (2007) 1182–1229; Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains (2008) Cambridge Univ. Press) establish the same result using representation theory. We are the first to handle k > 2 . We relate the PM RW to conjugacy-invariant RW s on the permutation group by introducing a "cycle structure" for PM s, then build on work of Berestycki, Schramm, Şengül and Zeitouni (Israel J. Math. 147 (2005) 221–243; Ann. Probab. 39 (2011) 1815–1843; Probab. Theory Related Fields 173 (2019) 1197–1241) on such RW s.
The random walk process underlies the description of a large number of real-world phenomena. Here we provide the study of random walk processes in time-varying networks in the regime of time-scale ...mixing, i.e., when the network connectivity pattern and the random walk process dynamics are unfolding on the same time scale. We consider a model for time-varying networks created from the activity potential of the nodes and derive solutions of the asymptotic behavior of random walks and the mean first passage time in undirected and directed networks. Our findings show striking differences with respect to the well-known results obtained in quenched and annealed networks, emphasizing the effects of dynamical connectivity patterns in the definition of proper strategies for search, retrieval, and diffusion processes in time-varying networks.
Efficient techniques to navigate networks with local information are fundamental to sample large-scale online social systems and to retrieve resources in peer-to-peer systems. Biased random walks, ...i.e. walks whose motion is biased on properties of neighbouring nodes, have been largely exploited to design smart local strategies to explore a network, for instance by constructing maximally mixing trajectories or by allowing an almost uniform sampling of the nodes. Here we introduce and study biased random walks on multiplex networks, graphs where the nodes are related through different types of links organised in distinct and interacting layers, and we provide analytical solutions for their long-time properties, including the stationary occupation probability distribution and the entropy rate. We focus on degree-biased random walks and distinguish between two classes of walks, namely those whose transition probability depends on a number of parameters which is extensive in the number of layers, and those whose motion depends on intrinsically multiplex properties of the neighbouring nodes. We analyse the effect of the structure of the multiplex network on the steady-state behaviour of the walkers, and we find that heterogeneous degree distributions as well as the presence of inter-layer degree correlations and edge overlap determine the extent to which a multiplex can be efficiently explored by a biased walk. Finally we show that, in real-world multiplex transportation networks, the trade-off between efficient navigation and resilience to link failure has resulted into systems whose diffusion properties are qualitatively different from those of appropriately randomised multiplex graphs. This fact suggests that multiplexity is an important ingredient to include in the modelling of real-world systems.
In a Quantum Walk (QW) the "walker" follows all possible paths at once through the principle of quantum superposition, differentiating itself from classical random walks where one random path is ...taken at a time. This facilitates the searching of problem solution spaces faster than with classical random walks, and holds promise for advances in dynamical quantum simulation, biological process modelling and quantum computation. Here we employ a versatile and scalable resonator configuration to realise quantum walks with bright classical light. We experimentally demonstrate the versatility of our approach by implementing a variety of QWs, all with the same experimental platform, while the use of a resonator allows for an arbitrary number of steps without scaling the number of optics. This paves the way for future QW implementations with spatial modes of light in free-space that are both versatile and scalable.
In this review, we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such ...as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spin models undergoing phase-ordering dynamics, diffusion equation, fluctuating interfaces, etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalizations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.
Recent years have witnessed an explosion of extensive geolocated datasets related to human movement, enabling scientists to quantitatively study individual and collective mobility patterns, and to ...generate models that can capture and reproduce the spatiotemporal structures and regularities in human trajectories. The study of human mobility is especially important for applications such as estimating migratory flows, traffic forecasting, urban planning, and epidemic modeling. In this survey, we review the approaches developed to reproduce various mobility patterns, with the main focus on recent developments. This review can be used both as an introduction to the fundamental modeling principles of human mobility, and as a collection of technical methods applicable to specific mobility-related problems. The review organizes the subject by differentiating between individual and population mobility and also between short-range and long-range mobility. Throughout the text the description of the theory is intertwined with real-world applications.
We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time n at an arbitrary value, encompassing in particular large deviation ...regimes on the boundary of the Cramér zone. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time n in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Łukasiewicz path of Bienaymé–Galton–Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy and Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian sphere.
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