Critical dimensionality for a special percolation problem

We consider a set of ideal chains, each with N beads (N >> 1) inscribed on a d-dimensional periodic lattice. Different chains are uncorrelated : thus any lattice site may belong to more than one chain ; two chains are said to be connected if they have at least one site in common. This defines a percolation problem (where the variable is the fraction c of occupied sites) physically related to a gelation process in polymers. For d > 4 the critical fraction c 0 is proportional to N-1 and the behaviour near c0 is of the mean field type. This simplification is due to the fact that c0 >> c*, where c* (∼ N1-d/2) is the concentration at which the chains begin to overlap. For d < 4 we expect c0 ∼c* and critical exponents not different from those of a site (or bond) percolation problem. We expect the same critical dimensionality to be maintained for the (more realistic) case of chains coupled by a repulsive interaction.