  Non-Gaussian diffusion is commonly considered as a result of fluctuating
diffusivity, which is correlated in time or in space or both. In this work, we
investigate the non-Gaussian diffusion in static disordered media via a
quenched trap model, where the diffusivity is spatially correlated. Several
unique effects due to quenched disorder are reported. We analytically estimate
the diffusion coefficient $D_{\text{dis}}$ and its fluctuation over samples of
finite size. We show a mechanism of population splitting in the non-Gaussian
diffusion. It results in a sharp peak in the distribution of displacement
$P(x,t)$ around $x=0$, that has frequently been observed in experiments. We
examine the fidelity of the coarse-grained diffusion map, which is
reconstructed from particle trajectories. Finally, we propose a procedure to
estimate the correlation length in static disordered environments, where the
information stored in the sample-to-sample fluctuation has been utilized.
