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We develop statistical enumeration methods for self-avoiding walks using apowerful sampling technique called the multicanonical Monte Carlo method. Usingthese methods, we estimate the numbers of the two dimensional N-stepself-avoiding walks up to N=256 with statistical errors. The developed methodsare based on statistical mechanical models of paths which include self-avoidingwalks. The criterion for selecting a suitable model for enumeratingself-avoiding walks is whether or not the configuration space of the modelincludes a set for which the number of the elements can be exactly counted. Wecall this set a scale fixing set. We selected the following two models whichsatisfy the criterion: the Go model for lattice proteins and the Domb-Joycemodel for generalized random walks. There is a contrast between these twomodels in the structures of the configuration space. The configuration space ofthe Go model is defined as the universal set of self-avoiding walks, and theset of the ground state conformation provides a scale fixing set. On the otherhand, the configuration space of the Domb-Joyce model is defined as theuniversal set of random walks which can be used as a scale fixing set, and theset of the ground state conformation is the same as the universal set ofself-avoiding walks. From the perspective of enumeration performance, weconclude that the Domb-Joyce model is the better of the two. The reason for theperformance difference is partly explained by the existence of the first-orderphase transition of the Go model.

Multicanonical simulation of the Domb-Joyce model and the Go model: new enumeration methods for self-avoiding walks