A self-modifying random walk on Q is deened from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of R having Hausdorr dimension less than 1. By generating function techniques we then calcul...

The Diestel-Leader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) Zq≀Z with respect to a natural generating set. For the “Simple random walk” (SRW) operator on the latter group, Grigorchuk and Żuk and Dicks and Schick have determined the spectrum and the (on-diago...

This paper establishes the first four moment expansions to the order o(a^−1) of S_{t_{a}}^{prime }/sqrt{t_{a}}, where S_{n}^{prime }=sum_{i=1}^{n}Y_{i} is a simple random walk with E(Yi) = 0, and ta is a stopping time given by t_{a}=inf left{ ngeq 1:n+S_{n}+zeta _{n}>aright}‎ where S_{n}=sum_{i=1}^{n}X_{i} is another simple random walk with E(Xi) = 0, and {zeta _{n},ngeq 1} is a sequence of ran...

Excited random walk is a model of a random walk on Z which, whenever it encounters a new vertex it receives a push toward a specific direction, call it the “right”, while when it reaches a vertex it “already knows”, it performs a simple random walk. This model has been suggested in [BW] and had since got lots of attention, see [V, Z]. The reason for the interest is that it is situated very natu...

We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...

Impairments in cell migration processes may cause various diseases, among which cancer cell metastasis, tumor angiogenesis, and the disability of immune cells to infiltrate into tumors are prominent ones. Mathematical modeling has been widely used to analyze the cell migration process. Cell migration is a complicated process and requires statistical methods such as random walk for proper analys...