Spiral self-avoiding walks on the triangular lattice

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Abstract
The authors study the behaviour of spiral self-avoiding walks on the triangular lattice. The spiral constraint simply says that no step in an anticlockwise direction may be taken. Imposing the additional constraint that steps may not deviate from the straight ahead direction by +pi/3 defines model I, previously solved by Joyce and Brak (1985) as well as by Lin (1985). If deviations of +2pi/3 are forbidden, this model is referred to as model II, while model III allows deviations of both +pi/3 and +2pi/3. One finds for both model II and model III that the number of n-step self-avoiding spirals is sn approximately c exp(2pi square root n)log(n/12)/n13/4 where c= phi2/768gamma5, phi(model II) approximately= 0.009, phi(model III) approximately= 0.16 and gamma = 1-12(log 2/pi)2. The confluent logarithm is an additional feature not present in the simpler case of the square lattice and model I triangular spiral self-avoiding walks. Use of two new results in the theory of partitions are made.

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