  The effect of the correlations in the diluteness pattern in the systems with
non-integral dimensionality, on $\nu=\frac{4}{5}$ superdiffusion process is
considered in this paper. These spatial correlations have proved to be very
effective in the critical phenomena. To simulate the particles motion in this
process, we employ the loop erased random walk (LERW). The spatial correlations
between imperfections (site-diluteness) also have been modeled by the Ising
model on a square lattice. It models the forbidden regions into which the
particles are not allowed to enter. The correlations are controlled by an
artificial temperature $T$. The trace of the walkers is shown to be
self-similar, whose fingerprint is the power-law behaviors. The detailed
analysis of the random walker's traces reveal that the (Ising-type)
correlations affect their geometrical properties. At the critical artificial
temperature $T_c$ we observe that the exponent of end-to-end distance $\nu$
becomes $0.807\pm 0.002$. The fractal dimension of the walker's trace is the
other geometrical quantity which scales inversely with the square root of the
correlation length of the Ising model, i.e. $D_f(T)-D_f(T_c)\sim
\zeta^{-\alpha}$ in which $\alpha=0.43\pm 0.05$. The winding angle test also
reveals that the traces are compatible with the Schramm-Loewner evolution
theory, and the diffusivity parameter for $T=T_c$ is $\kappa=1.89\pm 0.05$.
