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:
“Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don’t happen to have all the data. In mathematics we have all the data ... and yet we don’t understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.”
—Simone Weil (1909–1943)
“I shall have the veil withdrawn and be allowed to gaze unblinded on the narrow limits of my own possibilities.”
—Beatrice Potter Webb (1858–1943)
“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)