On the total number of distinct self-interacting self-avoiding walks on three-dimensional fractal structures

We have studied self-interacting self-avoiding walks (SAWs) situated on three-dimensional fractal structures, represented by a 3D Sierpinski gasket (SG) family of fractals. To this end, we have applied the exact renormalization group method (for the first three members of the SG fractal family, b = 2, 3 and 4) to calculate the critical exponents γE, γθ and γG (associated with the total number of distinct SAWs in extended phase, θ-chain and globular-collapsed polymer phase, respectively). To extend the obtained sequence of exact values for γE, we have applied the Monte Carlo renormalization group method for fractals with 2 ≤ b ≤ 40. Our results demonstrate that the critical exponent γE, as a function of b, monotonically increases and is always larger than the corresponding three-dimensional Euclidean value. On the other hand, γθ (for the values b = 2, 3 and 4), being smaller than the corresponding Euclidean value, decreases with b