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 limits of my language mean the limits of my world.”
—Ludwig Wittgenstein (1889–1951)
“Europe has what we do not have yet, a sense of the mysterious and inexorable limits of life, a sense, in a word, of tragedy. And we have what they sorely need: a sense of life’s possibilities.”
—James Baldwin (1924–1987)
“Whatever appeals to the imagination, by transcending the ordinary limits of human ability, wonderfully encourages and liberates us.”
—Ralph Waldo Emerson (1803–1882)