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 , while another variant, the kinetic prudent walk has been analyzed in detail in . 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.
"Scaling limit of the uniform prudent walk." Electron. J. Probab. 22 1 - 19, 2017. https://doi.org/10.1214/17-EJP87