Small-world phenomena and the statistics of linear polymer networks

Abstract

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