Exact solution for random walks on the triangular lattice with absorbing boundaries

Batchelor, Murray T.; Henry, Bryan R.

doi:10.1088/0305-4470/35/29/301

Cited by 10 publications

(8 citation statements)

References 12 publications

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“…Random walks, particularly on a lattice, have been extensively studied in the past [10] , [11] , [12] , [13] , [14] . Similar studies have also been used to analyze contact interactions [15] as stochastic processes, to better understand epidemic spread [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] .…”

Section: Monte Carlo Simulationsmentioning

confidence: 99%

A random walk Monte Carlo simulation study of COVID-19-like infection spread

Triambak

Mahapatra

2021

Physica A: Statistical Mechanics and its Applications

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Recent analysis of early COVID-19 data from China showed that the number of confirmed cases followed a subexponential power-law increase, with a growth exponent of around 2.2 (Maier and Brockmann, 2020). The power-law behavior was attributed to a combination of effective containment and mitigation measures employed as well as behavioral changes by the population. In this work, we report a random walk Monte Carlo simulation study of proximity-based infection spread. Control interventions such as lockdown measures and mobility restrictions are incorporated in the simulations through a single parameter, the size of each step in the random walk process. The step size is taken to be a multiple of , which is the average separation between individuals. Three temporal growth regimes (quadratic, intermediate power-law and exponential) are shown to emerge naturally from our simulations. For , we get intermediate power-law growth exponents that are in general agreement with available data from China. On the other hand, we obtain a quadratic growth for smaller step sizes , while for large the growth is found to be exponential. We further performed a comparative case study of early fatality data (under varying levels of lockdown conditions) from three other countries, India, Brazil and South Africa. We show that reasonable agreement with these data can be obtained by incorporating small-world-like connections in our simulations.

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Section: Monte Carlo Simulationsmentioning

confidence: 99%

A random walk Monte Carlo simulation study of COVID-19-like infection spread

Triambak

Mahapatra

2021

Physica A: Statistical Mechanics and its Applications

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“…First, true physical systems are not infinite, so that explicit boundary conditions have often to be taken into account in order to properly describe situations in which confinement can be relevant. Second, exact solvable random walk problems in bounded domains are very rare, making this theoretical field an important problem in its own right [8,9,10,11,12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

First-exit times and residence times for discrete random walks on finite lattices

2005

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In this paper, we derive explicit formulas for the surface averaged first-exit time of a discrete random walk on a finite lattice. We consider a wide class of random walks and lattices, including random walks in a nontrivial potential landscape. We also compute quantities of interest for modeling surface reactions and other dynamic processes, such as the residence time in a subvolume, the joint residence time of several particles, and the number of hits on a reflecting surface.

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“…The exact calculation of the importance of the state (q 1; 0,0) is not possible, see e.g., Batchelor and Henry (2002). To study it we will make the following assumption: "In a three-queue tándem Jackson network with loads p, > p 3 , if the initial system state is (q 1; 0,1) and the number of customers in the third queue reaches L before the third queue empties, q^ customers will be in the third queue when the first one becomes empty".…”

Section: Three-queue Tándem Jackson Networkmentioning

confidence: 99%

Importance functions for restart simulation of general Jackson networks

Villén-Altamirano

2010

European Journal of Operational Research

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RESTART is an accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. These regions are defined by means of a function of the system state called the importance function. Guidelines for obtaining suitable importance functions and formulas for the importance function of two-stage networks were provided in previous papers. In this paper, we obtain effective importance functions for RESTART simulation of Jackson networks where the rare set is defined as the number of customers in a particular ('target') node exceeding a predefined threshold. Although some rough approximations and assumptions are used to derive the formulas of the importance functions, they are good enough to estímate accurately very low probabilities for different network topologies within short computational time.

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Exact solution for random walks on the triangular lattice with absorbing boundaries

Abstract: Abstract. The problem of a random walk on a finite triangular lattice with a single interior source point and zig-zag absorbing boundaries is solved exactly. This problem has been previously considered intractable.

Cited by 10 publications

References 12 publications

A random walk Monte Carlo simulation study of COVID-19-like infection spread

A random walk Monte Carlo simulation study of COVID-19-like infection spread

First-exit times and residence times for discrete random walks on finite lattices

Importance functions for restart simulation of general Jackson networks

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