Limits
Consider the uniform measure on -step self-avoiding walks in the full plane. It is currently unknown whether the limit of the uniform measure as goes to infinity induces a measure on infinite full-plane walks. However, Harry Kesten has shown that such a measure exists for self-avoiding walks in the half-plane. One important question involving self-avoiding walks is the existence and conformal invariance of the scaling limit, that is, the limit as the length of the walk goes to infinity and the mesh of the lattice goes to zero. The scaling limit of the self-avoiding walk is conjectured to be described by Schramm–Loewner evolution with parameter .
Read more about this topic: Self-avoiding Walk
Famous quotes containing the word limits:
“In a virtuous action, I properly am; in a virtuous act, I add to the world; I plant into deserts conquered from Chaos and Nothing, and see the darkness receding on the limits of the horizon.”
—Ralph Waldo Emerson (1803–1882)
“Once and for all, there are many things I choose not to know.—Wisdom sets limits even to knowledge.”
—Friedrich Nietzsche (1844–1900)
“[I support] term limits for career politicians and the death penalty for career politicians.”
—William Frist (b. 1952)