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New Trends in Random Walks
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New Trends in Random Walks

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 May 2021) | Viewed by 23103

Special Issue Editor

Dr. Miquel Montero

E-Mail Website
Guest Editor
Departament de Física de la Matèria Condensada and Institute of Complex Systems (UBICS), Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
Interests: stochastic processes; mathematical finance; time-series analysis; game theory; probability and statistics; classical and quantum information theory

Special Issue Information

Dear Colleagues,

Random Walks are among the most recurrent stochastic processes, which appear in problems that come from many branches of science, not just physics. When present, Random Walks play different roles ranging from the modeling of the evolution financial asset prices and valuating their derivatives to the design of new search strategies in biological, technological or social contexts.

A key factor in understanding their success is the ability of Random Walks to extract valuable information from complex environments in a relatively simple way: through the erratic navigation of the graph representing the system by selecting from the nearest neighboring sites, with probabilities chosen uniformly. In recent years, however, there has been an increasing number of new revisions of this topic where the process behaves in a heterogeneous way, and these one-step probabilities may vary in space and time. The origin and nature of this lack of homogeneity is also diverse: Sometimes, it can be connected with the global properties of the graph, such as in the maximal entropy random walk, with the existence of an embedded metric space or with the presence of mechanism of aging, just to name a few.  

The purpose of this Special Issue is to provide an overview of the latest developments in Random Walks with a heterogeneous behavior. We encourage those researchers working in this kind of processes to send papers with novel contributions with a special emphasis on the interdisciplinary applications that may attract the interests of a wider audience. Although the above examples come from the scope of classical systems, quantum models are also welcome.

Dr. Miquel Montero
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • anomalous diffusion
  • biased processes
  • disordered systems
  • equilibrium
  • ergodicity
  • first-time events
  • localization
  • optimality
  • multilayer networks
  • search strategies

Published Papers (7 papers)

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Research

14 pages, 321 KiB  
Article
Mean Hitting Time for Random Walks on a Class of Sparse Networks
by Jing Su, Xiaomin Wang and Bing Yao
Entropy 2022, 24(1), 34; https://doi.org/10.3390/e24010034 - 24 Dec 2021
Cited by 1 | Viewed by 2082
Abstract
For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network [...] Read more.
For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks G(t) in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that G(t) has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of G(t), and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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13 pages, 328 KiB  
Article
A Semi-Deterministic Random Walk with Resetting
by Javier Villarroel, Miquel Montero and Juan Antonio Vega
Entropy 2021, 23(7), 825; https://doi.org/10.3390/e23070825 - 28 Jun 2021
Cited by 4 | Viewed by 2124
Abstract
We consider a discrete-time random walk (xt) which, at random times, is reset to the starting position and performs a deterministic motion between them. We show that the quantity [...] Read more.
We consider a discrete-time random walk (xt) which, at random times, is reset to the starting position and performs a deterministic motion between them. We show that the quantity Prxt+1=n+1|xt=n,n→∞ determines if the system is averse, neutral or inclined towards resetting. It also classifies the stationary distribution. Double barrier probabilities, first passage times and the distribution of the escape time from intervals are determined. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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Graphical abstract

19 pages, 2622 KiB  
Article
Random Walks with Invariant Loop Probabilities: Stereographic Random Walks
by Miquel Montero
Entropy 2021, 23(6), 729; https://doi.org/10.3390/e23060729 - 08 Jun 2021
Cited by 2 | Viewed by 2064
Abstract
Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. [...] Read more.
Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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24 pages, 2094 KiB  
Article
On (Non-)Monotonicity and Phase Diagram of Finitary Random Interlacement
by Zhenhao Cai, Yunfeng Xiong and Yuan Zhang
Entropy 2021, 23(1), 69; https://doi.org/10.3390/e23010069 - 04 Jan 2021
Cited by 5 | Viewed by 1620
Abstract
In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, for all d≥3, [...] Read more.
In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, for all d≥3, FRI is NOT stochastically monotone as fiber length increases. At the same time, numerical evidence still strongly supports the existence and uniqueness of a critical fiber length, which is estimated theoretically and numerically to be an inversely proportional function with respect to system intensity. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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34 pages, 852 KiB  
Article
Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing
by Viktor Stojkoski, Trifce Sandev, Lasko Basnarkov, Ljupco Kocarev and Ralf Metzler
Entropy 2020, 22(12), 1432; https://doi.org/10.3390/e22121432 - 18 Dec 2020
Cited by 37 | Viewed by 7237
Abstract
Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when [...] Read more.
Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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29 pages, 1413 KiB  
Article
A Continuous-Time Random Walk Extension of the Gillis Model
by Gaia Pozzoli, Mattia Radice, Manuele Onofri and Roberto Artuso
Entropy 2020, 22(12), 1431; https://doi.org/10.3390/e22121431 - 18 Dec 2020
Cited by 7 | Viewed by 2653
Abstract
We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic [...] Read more.
We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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22 pages, 479 KiB  
Article
Large Deviations for Continuous Time Random Walks
by Wanli Wang, Eli Barkai and Stanislav Burov
Entropy 2020, 22(6), 697; https://doi.org/10.3390/e22060697 - 22 Jun 2020
Cited by 26 | Viewed by 4132
Abstract
Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk [...] Read more.
Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory. Full article
(This article belongs to the Special Issue New Trends in Random Walks)
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