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New pattern theorems for square lattice self-avoiding walks and self-avoiding polygons

A general pattern theorem for weighted self-avoiding polygons (SAPs) and self-avoiding walks (SAWs) in is obtained. The pattern theorem for SAPs fits into the general framework of the pattern theorem for lattice clusters introduced by Madras (1999 Ann. Comb. 3 357–84). Note that, unlike other pattern theorems proved for SAPs, this pattern theorem does not rely on first establishing a relationship between SAPs and SAWs. These results are applied to obtain pattern theorems for self-interacting SAPs and self-interacting SAWs.