  Using stochastic conformal mapping techniques we study the patterns emerging
from Laplacian growth with a power-law decaying threshold for growth
$R_N^{-\gamma}$ (where $R_N$ is the radius of the $N-$ particle cluster). For
$\gamma > 1$ the growth pattern is in the same universality class as diffusion
limited aggregation (DLA) growth, while for $\gamma < 1$ the resulting patterns
have a lower fractal dimension $D(\gamma)$ than a DLA cluster due to the
enhancement of growth at the hot tips of the developing pattern. Our results
indicate that a pinning transition occurs at $\gamma = 1/2$, significantly
smaller than might be expected from the lower bound $\alpha_{min} \simeq 0.67$
of multifractal spectrum of DLA. This limiting case shows that the most
singular tips in the pruned cluster now correspond to those expected for a
purely one-dimensional line. Using multifractal analysis, analytic expressions
are established for $D(\gamma)$ both close to the breakdown of DLA universality
class, i.e., $\gamma \lesssim 1$, and close to the pinning transition, i.e.,
$\gamma \gtrsim 1/2$.
