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 $P(l)\sim l^{-\delta }$, in addition to the short range nearest neighbour connections, shows small-world behaviour for $0\le \delta <{\delta }_c$. In the most appropriate physical example of such a system, namely the linear polymer network, the exponent $\delta$ is related to the exponents of the corresponding n-vector model in the $n\to 0$ limit, and its value is less than ${\delta }_c$. 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 $\delta -q$ 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.

Minor corrections in text, Fig. 3 replaced

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