Scientific article
Open access

Scaling limit of the prudent walk

Published inElectronic communications in probability, vol. 15, no. 5, p. 44-58
Publication date2010

We describe the scaling limit of the nearest neighbour prudent walk on the square lattice, 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) = s_1 theta^+(3u/7) e_1 + s_2 theta^-(3u/7) e_2, where e_1, e_2 is the canonical basis, theta^+(t), resp. theta^-(t), is the time spent by a one-dimensional Brownian motion above, resp. below, 0 up to time t, and s_1, s_2 are two random signs. In particular, the asymptotic speed of the walk is well-defined in the L^1-norm and equals 3/7.

  • Prudent self-avoiding walk
  • Brownian motion
  • Scaling limit
  • Ballistic behaviour
  • Ageing
  • arxiv : math.PR
Citation (ISO format)
BEFFARA, Vincent, FRIEDLI, S., VELENIK, Yvan. Scaling limit of the prudent walk. In: Electronic communications in probability, 2010, vol. 15, n° 5, p. 44–58. doi: 10.1214/ecp.v15-1527
Main files (1)
Article (Accepted version)
ISSN of the journal1083-589X

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