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:
“This teaching is not practical in the sense in which the New Testament is. It is not always sound sense in practice. The Brahman never proposes courageously to assault evil, but patiently to starve it out. His active faculties are paralyzed by the idea of caste, of impassable limits of destiny and the tyranny of time.”
—Henry David Thoreau (1817–1862)
“As nature requires whirlwinds and cyclones to release its excessive force in a violent revolt against its own existence, so the spirit requires a demonic human being from time to time whose excessive strength rebels against the community of thought and the monotony of morality ... only by looking at those beyond its limits does humanity come to know its own utmost limits.”
—Stefan Zweig (18811942)
“You must confine yourself within the modest limits of order.”
—William Shakespeare (1564–1616)