Several years ago, Grimmett and Manolescu introduced an innovative technique that proved instrumental in establishing the universality of critical exponents within planar bond Bernoulli percolation models. This groundbreaking strategy, rooted in the star-triangle transformation, later found application in the broader realm of random cluster models. Notably, it was employed to demonstrate the rotation invariance of the model. In this presentation, we delve into the exploration of new avenues enabled by rotation invariance, aiming to extract fresh insights into critical random cluster models on Z^2.