Connectivity constant

The equations connecting constants of stability of initial and subsequent complexes with constants of replacement can be considered ... 

Direct connected, constant or variable speed, belt, gear, fluid coupling. 

Hammersley, JM, Percolation Processes. II. The Connective Constant, Proceedings of the Cambridge Philosophical Society 53, 642, 1957. 

This is due mainly to Hammersley6 and Kesten,7 who have been concerned with the total number of walks, C , and the number, w , which return to the origin at the nth step. They have established the existence of a number p for each lattice (which Hammersley terms the connective constant ) such that... 

The generator of the self-avoiding walk statistics or the distribution function Gjv(r), which represents the number of A-stepped SAW configurations with end-to-end distcince r, is not Gaussian as in the case of random walk (for random walk GN(r) exp[—r /A]). From the distribution function GAr(r), one can obtain the asymptotic behaviour of various moments. A brief summary is as follows The statistics of SAWs are characterised by the connectivity constant fi (Gn = M A" ), which is nonuniversal and depends... 

This fixed point is reached if we start with x = Xc = 0.27166, which corresponds to the connectivity constant /r = 3.68107. The linearization of recursion relations about this fixed point yields only one eigenvalue, Ai = 3.4965, higher than unity giving v = 0.5537 and a = 0.5686. 

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Variation of connectivity constant [fi) with 1//V for SAW on square lattice. The linear extrapolation gives [i = 2.636 which is quite close to the best estimated value... 

The fixed-length ensembles are best suited for studying the critical exponents V and 2A4 - 7, while the variable-length ensembles are best suited for studying the connective constant fj, and the critical exponents agi g (fixed-endpoint) or 7 (free-endpoint). Physically, the free-endpoint ensembles correspond to linear polymers, while the fixed-endpoint ensembles with x = 1 correspond to ring polymers,... 

Table 2.1 Connective constant /x and attrition constants A and X for simple (hyper- )cubic lattices in dimensions 2

To obtain the connective constant fjL, use the shthering-tortoise algorithm or the incomplete-enumeration algorithm. 

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