Prudent walk in dimension six and higher

TL;DR

This paper proves that high-dimensional prudent self-avoiding walks converge to Brownian motion under diffusive scaling, revealing a higher upper critical dimension due to strong self-avoidance constraints.

Contribution

It establishes the convergence of high-dimensional prudent walks to Brownian motion and identifies the upper critical dimension as five, which is higher than classical self-avoiding walks.

Findings

Prudent walk converges to Brownian motion in high dimensions.

Upper critical dimension for prudent walk is five.

Strong self-avoidance constraint influences critical dimension.

Abstract

We study the high-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. We prove that the prudent self-avoiding walk converges to Brownian motion under diffusive scaling if the dimension is large enough. The same result is true for weakly prudent walk in dimension d>5. A challenging property of the high-dimensional prudent walk is the presence of an infinite-range self-avoidance constraint. Interestingly, as a consequence of such a strong self-avoidance constraint, the upper critical dimension of the prudent walk is five, and thus greater than for the classical self-avoiding walk.