Scaling limit of the uniform prudent walk

Nicolas Pétrélis (1) , Rongfeng Sun (2) , Torri Niccolo (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 NUS - National University of Singapore

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-Mélou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the 2-dimensional uniform prudent walk is ballistic and follows one of the 4 diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.

Mots clés: self-avoiding walk; prudent walk; scaling limits

Domaines: Mathématiques [math]; Probabilités [math.PR]

Nicolas Pétrélis, Rongfeng Sun, Torri Niccolo. Scaling limit of the uniform prudent walk. Electronic Journal of Probability, 2017, 22 (66), pp.1 - 19. ⟨10.1214/17-EJP87⟩. ⟨hal-01470125v2⟩