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:
“It is of the essence of imaginative culture that it transcends the limits both of the naturally possible and of the morally acceptable.”
—Northrop Frye (b. 1912)
“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)
“This teaching is not practical in the sense in which the New Testament is. It is not always sound sense in practice. The Brahman never proposes courageously to assault evil, but patiently to starve it out. His active faculties are paralyzed by the idea of caste, of impassable limits of destiny and the tyranny of time.”
—Henry David Thoreau (1817–1862)