  The raise and peel model (RPM) is a nonlocal stochastic model describing the
space and time fluctuations of an evolving one dimensional interface. Its
relevant parameter $u$ is the ratio between the rates of local adsorption and
nonlocal desorption processes (avalanches) processes. The model at $u=1$ give
us the first example of a conformally invariant stochastic model. For small
values $u<u_0$ the model is known to be noncritical, while for $u>u_0$ it is
critical.
  By calculating the structure function of the height profiles in the
reciprocal space we confirm with good precision that indeed $u_0=1$. We
establish that at the conformal invariant point $u=1$ the RPM has a roughness
transition with dynamical and roughness critical exponents $z=1$ and
$\alpha=0$, respectively. For $u>1$ the model is critical with an $u$-dependent
dynamical critical exponent $z(u)$ that tends towards zero as $u\to \infty$.
However at $1/u=0$ the RPM is exactly mapped into the totally asymmetric
exclusion problem (TASEP). This last model is known to be noncritical
(critical) for open (periodic) boundary conditions. Our studies indicate that
the RPM as $u \to \infty$, due to its nonlocal dynamics processes, has the same
large-distance physics no matter what boundary condition we chose. For $u>1$,
our analysis show that differently from previous predictions, the region is
composed by two distinct critical phases. For $u\leq u < u_c\approx 40$ the
height profiles are rough ($\alpha = \alpha(u) >0$), and for $u>u_c$ the height
profiles are flat at large distances ($\alpha = \alpha(u) <0$). We also
observed that in both critical phases ($u>1$) the RPM at short length scales,
has an effective behavior in the Kardar-Parisi-Zhang (KPZ) critical
universality class, that is not the true behavior of the system at large length
scales.
