The central concern of this thesis is the study of the Russo-Seymour-Welsh (RSW) theory. The first contribution of this thesis is a macroscopic decorrelation result for uniform spanning trees (USTs) on random planar graphs based on the RSW assumption. A similar result was established on a fixed graph in [BLR20, Theorem 4.21]. We extend this result to USTs on random graphs. In particular, we show that a similar coupling can be obtained for a collection of graphs, which has a high probability. This is the key missing step in the application of the proof strategy in [BLR20] for random graphs, which established the scaling limits of height function of dimer model to a Gaussian free field on a fairly general class of fixed graphs. The second contribution of this thesis is the RSW type results for random walks on two concrete and natural examples: the unique infinite cluster of supercritical bond percolation in $\Z^2$ and the Poisson-Delaunay triangulation in $\R^2$. We show that random walks crossing a rectangle without exiting occurs with a stretched exponentially high in the scale. The main tool used in the proof is heat kernel estimates for random walks on the supercritical bond percolation. The proof of RSW for bond percolation is a quick application of a combination of Barlow's results. However, we cannot apply Barlow's results for the Delaunay triangulation directly since there is no uniform bound on degree. A key input is a quantitative isoperimetric inequality for the Delaunay triangulation, which we consider to be another novel contribution of this thesis.