We consider the continuous time version of the 'true' or 'myopic' self-avoiding random walk with site repulsion in 1d. The Ray – Knight-type method which was applied in [15] to the discrete time and edge repulsion case, is applicable to this model with some modifications. We present a limit theorem for the local time of the walk and a local limit theorem for the displacement.

‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t. Let Γ(t) be the subgraph of G induced by the vacant set of the walk at step t. Similarly, let Γ̂(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously est...

The random walk on a discrete lattice has been analysed in completely different fields such as chemistry [1, 2], ecology [3, 4], and general physics [5, 6]. The general idea has been to insert a diffusion point at the centre of a 1D, 2D, or 3D discrete lattice and to follow the evolution from an analytical or numerical [7–9] point of view. Here we will start by analysing the stationary state of...

Random walks can be used to solve the Dirichlet problem – the boundary value problem for harmonic functions. We begin by constructing the random walk in Z and showing some of its properties. Later, we introduce and examine harmonic functions in Z in order to set up the discrete Dirichlet problem. Finally, we solve the Dirichlet problem using random walks. Throughout the paper, we discuss connec...

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the “generation gap” process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the...

We consider a large class of random walks on the discrete circle Z=(n), deened in terms of a piecewise Lipschitz function, and motivated by the \generation gap" process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z=(n) with a random walk on the ...

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...