Equidistribution of graphs of holomorphic correspondences

Let \(X\) be a compact Riemann surface. Let \(f\) be a holomorphic self-correspondence of \(X\) with dynamical degrees \(d_1\) and \(d_2\). Assume that \(d_1\neq d_2\) or \(f\) is non-weakly modular. We show that the graphs of the iterates \(f^n\) of \(f\) are equidistributed exponentially fast with respect to a positive closed current in \(X\times X\).