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:
“[I support] term limits for career politicians and the death penalty for career politicians.”
—William Frist (b. 1952)
“And when discipline is concerned, the parent who has to make it to the end of an eighteen-hour day—who works at a job and then takes on a second shift with the kids every night—is much more likely to adopt the survivor’s motto: “If it works, I’ll use it.” From this perspective, dads who are even slightly less involved and emphasize firm limits or character- building might as well be talking a foreign language. They just don’t get it.”
—Ron Taffel (20th century)
“I fear chiefly lest my expression may not be extra-vagant enough, may not wander far enough beyond the narrow limits of my daily experience, so as to be adequate to the truth of which I have been convinced.”
—Henry David Thoreau (1817–1862)