We consider discrete random fractal surfaces with negative Hurst
exponent H<0H<0.
A random colouring of the lattice is provided by activating the sites at
which the surface height is greater than a given level
hh.
The set of activated sites is usually denoted as the excursion set. The
connected components of this set, the level clusters, define a
one-parameter (HH)
family of percolation models with long-range correlation in the site
occupation. The level clusters percolate at a finite value
h=h_ch=hc
and for H\leq-\frac{3}{4}H≤−34
the phase transition is expected to remain in the same universality
class of the pure (i.e. uncorrelated) percolation. For
-\frac{3}{4}<H< 0−34<H<0
instead, there is a line of critical points with continously varying
exponents. The universality class of these points, in particular
concerning the conformal invariance of the level clusters, is poorly
understood. By combining the Conformal Field Theory and the numerical
approach, we provide new insights on these phases. We focus on the
connectivity function, defined as the probability that two sites belong
to the same level cluster. In our simulations, the surfaces are defined
on a lattice torus of size M\times NM×N.
We show that the topological effects on the connectivity function make
manifest the conformal invariance for all the critical line
H<0H<0.
In particular, exploiting the anisotropy of the rectangular torus
(M\neq NM≠N),
we directly test the presence of the two components of the traceless
stress-energy tensor. Moreover, we probe the spectrum and the structure
constants of the underlying Conformal Field Theory. Finally, we observed
that the corrections to the scaling clearly point out a breaking of
integrability moving from the pure percolation point to the long-range
correlated one.
The Structure of Connectivity Functions
