First-exit times for a two-dimensional random walk on the L-lattice
One of the main topic in the theory of stochastic processes is constituted by random walks on graphs and networks, since they are used as simple models to study a variety of mathematical problems. Indeed, problems like diffusions on manifolds, harmonic analysis, infinite graph theory, group theory, etc. are addressed with the use of specific random walks on networks (see, e.g. [3] and related references). Models of random walks have been used to describe a variety of transport processes, also in physical systems. For instance in [1] and [2] the Manhattan lattice and the L-lattice are used to study the quantum and classical localization and ordinary integer quantum Hall transition, respectively. The study of random walks on lattices concerns mainly the analysis of the characteristics of the states of the related Markov chain and the asymptotic behavior, see for example [4]. Our aim is to study a two-dimensional random walk on the L-lattice, focusing on the state probabilities and on some first-passage-time (FPT) problems. The considered L-lattice is an unbounded square lattice consisting of directed edges and nodes, such that a particle that starts at a vertex (x,y) moves to an adjacent vertex following an appropriate transition probability. In this framework, we study the random walk on the L-lattice defined according to the following rules. Let V0 (resp., V1) be the set of points (x, y) such that x+y is even (resp., odd), for x, y ∈ Z. Moreover, if the particle is located in a vertex of V0, then it can reach one of the two adjacent positions on the right (with probability q) or on the left (with probability 1−q); if the particle is located in a vertex of the set V1, then it can reach one of the two adjacent positions on the top (with probability p) or on the bottom (with probability 1 − p). We firstly determine the probability generating functions related to the sets V0 and V1, the transition probabilities and the main moments of the described random walk. Then, we investigate the FPT problems of the random walk through certain straight line boundaries. We also focus on the `taboo probabilities' concerning transitions on the nodes of the L-lattice that avoid the boundary. We determine the taboo probabilities and, thanks to suitable symmetry properties, the FPT probabilities, the crossing probabilities and the FPT moments. For other kinds of boundaries we use a simulating procedure in order to address the FPT problem. References [1] E.J. Beamond, A.L. Owczarek, J. Cardy, Quantum and classical localization and the Manhattan lattice, J. Phys. A: Math. Gen. 36 (2003), 10251-10267. [2] N.R. Beaton, M. Holmes, The mean square displacement of random walk on the Manhattan lattice, Statistics & Probability Letters 193 (2023), 109706. [3] M. Campanino, D. Petritis, Type transition of simple random walks on randomly directed regular lattices. J. Appl. Prob. 51 (2014), 1065-1080. [4] G. Lawler,V. Limic, Random Walk: A Modern Introduction, Cambridge: Cambridge University Press, 2010.
Area: CS54 - Random motions and first passage times (Alessandra Meoli and Costantino Ricciuti)
Keywords: first-passage-time, random walk, lattice
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