Folding of the triangular lattice with quenched random bending rigidity

We study the problem of folding of the regular triangular lattice in the presence of a quenched random bending rigidity ±K and a magnetic field h (conjugate to the local normal vectors to the triangles). The randomness in the bending energy can be understood as arising from a prior marking of the lattice with quenched creases on which folds are favored. We consider three types of quenched randomness:m(i) a 'physical' randomness where the creases arise from some prior random folding; (ii) a Mattis-like randomness where creases are domain walls of some quenched spin system; (iii) an Edwards-Anderson-like randomness where the bending energy is ±K at random, independently on each bond. The corresponding (K,h) phase diagrams are determined in the hexagon approximation of the cluster variation method. Depending on the type of randomness, the system shows essentially different behaviors.