Small-world phenomena and the statistics of linear polymers

A regular lattice in which the sites can have long-range connections at a distance l with a probability P(l)∼l-δ, in addition to the short-range nearest neighbour connections, shows small-world behaviour for 0≤ δ<δc. In the most appropriate physical example of such a system, namely, the linear polymer network, the exponent δ is related to the exponents of the corresponding n-vector model in the n→0 limit, and its value is less than δc. Still, the polymer networks do not show small-world behaviour. Here, we show that this is due to a (small value) constraint on the number, q, of long-range connections per monomer in the network. In the general δ-q space, we obtain a phase boundary separating regions with and without small-world behaviour, and show that the polymer network falls marginally in the regular lattice region. (author)