Small-world phenomena and the statistics of linear polymers

Abstract

A regular lattice in which the sites can have long-range connections at a distance l with a probabilty P(l) ˜ l ^-δ, in addition to the short-range nearest neighbour connections, shows small-world behaviour for 0 ≤ δ < δ_c. 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 δ_c. 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.