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arXiv · 1702.04915

Scaling limit of the uniform prudent walk

Nicolas Pétrélis·Rongfeng Sun·Niccolò Torri
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Abstract

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.

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Nicolas Pétrélis, Rongfeng Sun, Niccolò Torri. 2017-09-07. Scaling limit of the uniform prudent walk. https://doi.org/10.1214/17-ejp87

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