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:
“Yet shall he mount, and keep his distant way
Beyond the limits of a vulgar fate:
Beneath the Good how far—but far above the Great.”
—Thomas Gray (1716–1771)
““... Ain’t it a caution to us not to fix
No limits to what rose in rubbing sticks
On fire to scare away the pterodix
When man first lived in caves along the creeks?”
“Marvelous world in nineteen-twenty-six.””
—Robert Frost (1874–1963)
“I shall have the veil withdrawn and be allowed to gaze unblinded on the narrow limits of my own possibilities.”
—Beatrice Potter Webb (1858–1943)