Abstract

Scaling Limit of the Prudent Walk V. Beffara, S. Friedli and Y. Velenik Electron. Commun. Probab. 15, 44-58 (2010). We describe the scaling limit of the nearest neighbour prudent walk on ${\bf Z}^2$, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process $Z_u = \int_0^{3u/7} \bigl( \sigma_1 1_{\{W(s)\geq 0\}} \vec{e}_1 + \sigma_2 1_{\{W(s)< 0\}} \vec{e}_2 \bigr){\rm d} s,\quad u\in[0,1],$ where $W$ is the one-dimensional Brownian motion and $\sigma_1,\sigma_2$ two random signs. In particular, the asymptotic speed of the walk is well-defined in the $L^1$-norm and equals 3/7. Key words: Prudent Self-Avoiding Walk, Scaling Limit, Ballistic Behaviour, Ageing. Files: PDF file, Published version, bibtex