Overall properties of a discrete membrane with randomly distributed defects

A prototype for variational percolation problems with surface energies is considered: the description of the macroscopic properties of a (two-dimensional) discrete membrane with randomly distributed defects in the spirit of the weak membrane model of Blake and Zisserman (quadratic springs that may break at a critical length of the elongation). After introducing energies depending on suitable independent identically distributed random variables, this is done by exhibiting an equivalent continuum energy computed using Delta-convergence, geometric measure theory, and percolation arguments. We show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the Dirichlet integral, while above that threshold an additional (Griffith) fracture term appears in the energy, which depends only on the defect probability through the chemical distance on the "weak cluster of defects".