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:
“Now rest in peace, our patriot band;
Though far from nature’s limits thrown,
We trust they find a happier land,
A brighter sunshine of their own.”
—Philip Freneau (1752–1832)
“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)
“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)