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:
“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)
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The one-foot stars were couriers of its death
To the wild limits of its habitation.
These were not tepid stars of torpid places
But bravest at midnight and in lonely spaces,
They looked back at Hans’ look with savage faces.”
—Wallace Stevens (1879–1955)
“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)