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)
“[I support] term limits for career politicians and the death penalty for career politicians.”
—William Frist (b. 1952)
“The myth of independence from the mother is abandoned in mid- life as women learn new routes around the mother—both the mother without and the mother within. A mid-life daughter may reengage with a mother or put new controls on care and set limits to love. But whatever she does, her child’s history is never finished.”
—Terri Apter (20th century)