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:
“Mysticism and exaggeration go together. A mystic must not fear ridicule if he is to push all the way to the limits of humility or the limits of delight.”
—Milan Kundera (b. 1929)
“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)
“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)