5 research outputs found
Critical behavior of loops and biconnected clusters on fractals of dimension d < 2
We solve the O(n) model, defined in terms of self- and mutually avoiding
loops coexisting with voids, on a 3-simplex fractal lattice, using an exact
real space renormalization group technique. As the density of voids is
decreased, the model shows a critical point, and for even lower densities of
voids, there is a dense phase showing power-law correlations, with critical
exponents that depend on n, but are independent of density. At n=-2 on the
dilute branch, a trivalent vertex defect acts as a marginal perturbation. We
define a model of biconnected clusters which allows for a finite density of
such vertices. As n is varied, we get a line of critical points of this
generalized model, emanating from the point of marginality in the original loop
model. We also study another perturbation of adding local bending rigidity to
the loop model, and find that it does not affect the universality class.Comment: 14 pages,10 figure
Universality classes of dense polymers and conformal sigma models
In the usual statistical model of a dense polymer (a single space-filling
loop on a lattice) in two dimensions the loop does not cross itself. We modify
this by including intersections in which {\em three} lines can cross at the
same point, with some statistical weight w per crossing. We show that our model
describes a line of critical theories with continuously-varying exponents
depending on w, described by a conformally-invariant non-linear sigma model
with varying coupling constant g_\sigma^2 >0. For the boundary critical
behavior, or the model defined in a strip, we propose an exact formula for the
\ell-leg exponents, h_\ell=g_\sigma^2 \ell(\ell-2)/8, which is shown
numerically to hold very well.Comment: 5 page