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:
“The great ship, Balayne, lay frozen in the sea.
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)
“Caring for children is a dance between setting appropriate limits as caretakers and avoiding unnecessary power struggles that result in unhappiness.”
—Charlotte Davis Kasl (20th century)
“Predatory capitalism created a complex industrial system and an advanced technology; it permitted a considerable extension of democratic practice and fostered certain liberal values, but within limits that are now being pressed and must be overcome. It is not a fit system for the mid- twentieth century.”
—Noam Chomsky (b. 1928)