Scaling function and universal amplitude combinations for self-avoiding polygons

Abstract

We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted self-avoiding polygons may satisfy a q-algebraic functional equation.

Exact scaling functions for self-avoiding loops and branched polymers

“…Thus, apart from non-universal constants, G According to (2), the scaling function F (s) has singularities at the zeros of the Airy function, which lie on the negative real axis. The closest to x = 0 lies at x c (g) = x c + (2.388 .…”

On the area under a continuous time Brownian motion till its first-passage time

“…[2], it is known that the area-perimeter generating function has a universal scaling form about a tri-critical point (this form being, essentially, the logarithmic derivative of the Airy function) [2,3]. There is powerful evidence that the same basic scaling form also describes the behaviour of all rooted self-avoiding polygons [4]. It is only for staircase polygons, however, that there exists a rigorous proof of this result starting from the area-perimeter generating function itself, and even then the analysis is far from trivial [5].…”

Asymptotic Behavior of Inflated Lattice Polygons

“…At p = 0, the problem reduces to that of the enumeration of self-avoiding polygons. The scaling function describing the scaling behavior (for p < 0) near the tricritical point * Electronic address: mithun@imsc.res.in † Electronic address: menon@imsc.res.in ‡ Electronic address: rrajesh@imsc.res.in p = 0 and µ = κ −1 , where κ is the growth constant for self-avoiding polygons, is also known exactly [14,15,16].…”