Statistics of nested spiral self-avoiding loops: exact results on the square and triangular lattices

The statistics of nested spiral, self-avoiding loops, which is closely related to the partition of integers into decreasing parts, has been studied on the square and triangular lattices. The number of configurations with N steps is c_N approximately=(2/24)N^-32/ exp( pi 2/3 N¹2/) and their average size X_N approximately=(1/2 pi )3/2 N¹2/ 1n N to leading order on the square lattice while the corresponding values for the triangular lattice are c_N approximately=(3³4//16) N^-54/ exp(( pi /3) N¹2/) and X_N approximately=1/( pi 3)N¹2/ 1n N.