Independently orient each horizontal and vertical line of $\mathbb{Z}^2$ by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. Redner (1989) introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like $n^{2/3}$. We prove that the walk is transient almost surely. The proof is a short variational argument.
We consider a competition between two independent random walks on a cycle of length $N$. Each vertex is claimed by the walker that visits it first, and remains claimed thereafter. We prove that if the initial distance between the walkers is $d$, then the expected number of edges whose endpoints are claimed by different...
S. Chatterjee, Nadya Nabahi, Grigory Terlov· 1 citation
We study the superdiffusive behavior of random walks on the randomly oriented Manhattan lattice, i.e., the $d$-dimensional integer lattice $\mathbb{Z}^d$ where each axis-aligned line is independently assigned a random direction (forward or backward) with equal probability. The walker takes nearest-neighbor steps, choos...
In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishn...
We consider a once-excited random walk on $\mathbb{Z}^d$, $d \ge 2$, where the walk on its first visit has a bias of constant strength $\beta>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape...
We study the limit shape of the range of the random walk excited to the center on $\mathbb Z^d$, with $d\geq2$. A cookie is placed at each nonzero vertex. On its first visit to such a vertex, the walker consumes the cookie and takes its next nearest-neighbor step with a bias toward the origin. On a visit to the origin,...
In this paper we consider random walks moving in a dynamic random environment that is completely resampled with probability $p$ after each step of the random walker. This resampling yields a straightforward regeneration structure and limit theorems, such as a law of large numbers with a limiting speed $v(p)$ that depen...
Raphael Martin, J. Nagel· 0 citations
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