A shape theorem for a radially excited random walk in dimensions $d \ge 2$

Abstract

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 theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.