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)
“The element running through entire nature, which we popularly call Fate, is known to us as limitation. Whatever limits us, we call Fate.”
—Ralph Waldo Emerson (1803–1882)
“The limits of prudence: one cannot jump out of a burning building gradually.”
—Mason Cooley (b. 1927)