  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.
