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Overall properties of a discrete membrane with randomly distributed defects

In this paper we study the problem of the description of the overall behaviour of a discrete membrane with defects. Our model mixes two extreme behaviours and is set in a variational framework. One extreme bahaviour is that of a ‘strong membrane’, consisting just in the finite-difference approximation of the Dirichlet integral. The defects are introduced as nonlinear pair interactions as in Blake and Zisserman’s weak membrane model in Computer Vision [3]. This model can be translated into a simple non-linear finite-difference scheme (see Chambolle [10]) on a two-dimensional (in general, d-dimensional) grid. Its underlying discrete energy is of the form