Nina Javerzat, Sebastian Grijalva, Alberto Rosso, Raoul Santachiara
SciPost Phys. 9, 050 (2020) ·
published 12 October 2020

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We consider discrete random fractal surfaces with negative Hurst exponent
$H<0$. A random colouring of the lattice is provided by activating the sites at
which the surface height is greater than a given level $h$. The set of
activated sites is usually denoted as the excursion set. The connected
components of this set, the level clusters, define a oneparameter ($H$) family
of percolation models with longrange correlation in the site occupation. The
level clusters percolate at a finite value $h=h_c$ and for $H\leq\frac{3}{4}$
the phase transition is expected to remain in the same universality class of
the pure (i.e. uncorrelated) percolation. For $\frac{3}{4}<H< 0$ 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 $M\times N$. We show that the
topological effects on the connectivity function make manifest the conformal
invariance for all the critical line $H<0$. In particular, exploiting the
anisotropy of the rectangular torus ($M\neq N$), we directly test the presence
of the two components of the traceless stressenergy 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
longrange correlated one.