Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $\delta$, the discretized functionals $\mathrm{\Gamma }$-converge to the Mumford-Shah functional only if $\delta \ll \epsilon$, $\epsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $\mathrm{\Gamma }$-convergence result also for $\delta \sim \epsilon$, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford-Shah functional.