Universality of the Gaussian Free Field in the Critical Ising Model on Isoradial Graphs

We establish the universality of the Gaussian Free Field (GFF) as the scaling limit of the height function associated with the critical Ising model on the class of isoradial graphs with Z-invariant weights. By exploiting the Fisher correspondence, we map the Ising model to a dimer model on a decorated graph and analyze the fluctuations of the associated height function. Utilizing the framework of discrete complex analysis on isoradial graphs, specifically the convergence of discrete holomorphic observables and the inverse Kasteleyn operator, we prove that the centered height function converges in distribution to the GFF with Dirichlet boundary conditions. This result extends the universality class of the GFF beyond the square lattice to a broad family of planar graphs, confirming the robustness of conformal invariance in 2D statistical mechanics.