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 cN ≃ ( √ 2/24)N−3/2 exp(π √ 2 3 N1/2) and their average sizeXN ≃ (1/2π) √ 3 2 N1/2 lnN to leading order on the square lattice while the corresponding values for the triangular lattice are cN ≃ (33/4/16)N−5/4 exp((π/ √ 3)N1/2) and XN ≃ 1/(π √ 3)N1/2 lnN . Some years ago, the number of N -step spiral self–avoiding loops have been calculated for the square and triangular lattices (Manna 1985, Lin et al 1986). These works followed the introduction of the spiral self–avoiding walk (Privman 1983) for which a lot of exact results were obtained by a succession of authors (Blöte and Hilhorst 1984, Whittington 1984, Gutmann and Wormald 1984, Joyce 1984, Guttmann and Hirschhorn 1984, Lin 1985, Joyce and Brak 1985, Lin and Liu 1986). While the number of spiral self–avoiding loops grows with N like a nonuniversal, i.e. lattice– dependent power, the number of spiral self–avoiding walks also behaves in an unusual way CN ≃ AN exp (λN) (1) where both θ and λ are lattice–dependent quantities. It follows that the asymptotic entropy per step decays as N instead of giving a constant like in the ordinary or directed self–avoiding walks. Following the work of Privman, a close connection between this problem and the theory of partitions of integers (Andrews 1976) was noticed (Derrida and Nadal 1984, Redner and de Arcangelis 1984, Klein et al 1984). In this letter we present some exact results concerning the statistics of nested spiral loops which are self– and mutually avoiding and piled up around a site chosen as origin. We study such spirals on the square lattice where on a loop only 90 turns in the same direction are allowed so that the loops are rectangular–shaped (figure 1) and on the triangular lattice where the restriction to 120 turns leads to triangles among which one only keeps those pointing up (figure 2). † Unité de Recherche associée au CNRS no 155 L1119 L1120 Letter to the Editor Figure 1. Nested spiral self–avoiding loops on the square lattice: with 90 turns in the same direction, rectangular–shaped loops are obtained. Each step is assigned a weight z and the size is X= ∑L k=1 Xk. The nested–loop configuration corresponds to four independent partitions of integers into decreasing parts numbered 1 to 4 and each partition is duplicated (heavy lines). Figure 2. Nested spiral self–avoiding loops on the triangular lattice: with 120 turns in the same direction, triangular–shaped loops are obtained. Each step is assigned a weight z and the size is X= ∑L k=1 Xk. The nested–loop configuration corresponds to three independent partitions of integers into decreasing parts numbered 1 to 3 and each partition is triplicated (heavy lines). Let us introduce the generating function GL(z, ω) = ∑ N,X cN (L,X)z e (2) for the number of configurations with N steps, L loops and size X= ∑L k=1 Xk where X is the distance from the origin to the Lth loop. On the square lattice, with the notations of figure 1, one may write G L (z, ω) = ∞ ∑