Latest Advances in Random Walks Dating Back to One Hundred Years

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 8490

Special Issue Editor

Department of Quantitative Methods, University of Pannonia, Veszpr em, Hungary
Interests: random walks; Markov chains; causality; diffusion; fractals; networks; mathematical models in computational neuroscience
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Random walks on graphs may be the simplest stochastic process. Thanks to their simplicity, they serve as a model in many areas beyond probability. Among others, they are the fundamental model of diffusion in physics. The first astonishing observation was made by György Pólya in 1921 about the recurrence of the random walk on d-dimensional lattice. Numerous great mathematicians have since then contributed to the development of its theory, such as Paul Erdős, Alfréd Rényi, A. Doverczky, A. Kakutani, W. Feller,Y.G. Sinai, R.L. Dobrushin, D. Aldous, S. Redner. G. Weiss, F. Den Hollander, H. Kesten, F. Solomon, W. Woess, B.D. Hughes, P.G. Doyle, J.L. Snell, S. Alexander, R.Orbach, R. Rammal, G. Toulouse, G.F. Lawler, N. Madras, G. Slade, L. Lovász, P. Révész, J. Norris, V.A. Kaimanovitch, A. Grigor’yan, L. Saloff-Coste, T. Coulhon, O. Zeitouni, G. Kozma, and Y. Peres, just to name some. Recent advances have widened the picture to sub- and superdiffusive phenomena, metastability, and stock price movements, again to name only a fraction of the options.

The aim of the present Special Issue is to provide a wide spectrum but at the same time deep insight into recent results with respect to random walk on its own, as well as its essential applications. We seek papers which provide new and striking insight into the nature of random walk or phenomena modelled by random walk. The emphasis is on the interesting phenomenon in the spirit of György Pólya, not on incremental technical advances. Survey papers might be accommodated in the Special Issue, one for each specific broad area such as RWs on graphs, RWs in random environments, RWs in physics, chemistry, biology, economics, etc.

András Telcs
Guest Editor

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Keywords

  • Random walks
  • Random walks on graphs
  • Random walks on fractals
  • Random walks in science

Published Papers (4 papers)

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30 pages, 5519 KiB  
Article
Phase-Type Distributions of Animal Trajectories with Random Walks
by Rodolfo Vera-Amaro, Mario E. Rivero-Ángeles and Alberto Luviano-Juárez
Mathematics 2023, 11(17), 3671; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173671 - 25 Aug 2023
Viewed by 599
Abstract
Animal monitoring systems often rely on expensive and challenging GPS-based systems to obtain accurate trajectories. However, an alternative approach is to generate synthetic trajectories that exhibit similar statistical properties to real trajectories. These synthetic trajectories can be used effectively in the design of [...] Read more.
Animal monitoring systems often rely on expensive and challenging GPS-based systems to obtain accurate trajectories. However, an alternative approach is to generate synthetic trajectories that exhibit similar statistical properties to real trajectories. These synthetic trajectories can be used effectively in the design of surveillance systems such as wireless sensor networks and drone-based techniques, which aid in data collection and the delineation of areas for animal conservation and reintroduction efforts. In this study, we propose a data generation method that utilizes simple phase-type distributions to produce synthetic animal trajectories. By employing probability distribution functions based on the exponential distribution, we achieve highly accurate approximations of the movement patterns of four distinct animal species. This approach significantly reduces processing time and complexity. The research primarily focuses on generating animal trajectories for four endangered species, comprising two terrestrial and two flying species, in order to demonstrate the efficacy of the proposed method. Full article
(This article belongs to the Special Issue Latest Advances in Random Walks Dating Back to One Hundred Years)
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20 pages, 367 KiB  
Article
Networks with Complex Weights: Green Function and Power Series
by Anna Muranova and Wolfgang Woess
Mathematics 2022, 10(5), 820; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050820 - 04 Mar 2022
Cited by 1 | Viewed by 1483
Abstract
We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the [...] Read more.
We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting. Full article
(This article belongs to the Special Issue Latest Advances in Random Walks Dating Back to One Hundred Years)
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20 pages, 597 KiB  
Article
A Simple Out-of-Sample Test of Predictability against the Random Walk Benchmark
by Pablo Pincheira, Nicolas Hardy and Andrea Bentancor
Mathematics 2022, 10(2), 228; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020228 - 12 Jan 2022
Cited by 5 | Viewed by 1343
Abstract
We show that a straightforward modification of a trading-based test for predictability displays interesting advantages over the Excess Profitability (EP) test proposed by Anatolyev and Gerco when testing the Driftless Random Walk Hypothesis. Our statistic is called the Straightforward Excess Profitability (SEP) test, [...] Read more.
We show that a straightforward modification of a trading-based test for predictability displays interesting advantages over the Excess Profitability (EP) test proposed by Anatolyev and Gerco when testing the Driftless Random Walk Hypothesis. Our statistic is called the Straightforward Excess Profitability (SEP) test, and it avoids the calculation of a term that under the null of no predictability should be zero but in practice may be sizable. In addition, our test does not require the strong assumption of independence used to derive the EP test. We claim that dependence is the rule and not the exception. We show via Monte Carlo simulations that the SEP test outperforms the EP test in terms of size and power. Finally, we illustrate the use of our test in an empirical application within the context of the commodity-currencies literature. Full article
(This article belongs to the Special Issue Latest Advances in Random Walks Dating Back to One Hundred Years)
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38 pages, 558 KiB  
Article
Current Trends in Random Walks on Random Lattices
by Jewgeni H. Dshalalow and Ryan T. White
Mathematics 2021, 9(10), 1148; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101148 - 19 May 2021
Cited by 7 | Viewed by 4006
Abstract
In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice [...] Read more.
In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing. Full article
(This article belongs to the Special Issue Latest Advances in Random Walks Dating Back to One Hundred Years)
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