Excluded volume and hyperscaling in polymeric systems

A dense system of mutually avoiding polymers, obeying a quenched power-law mass distribution, n(m) ∼ m-τ (with low and high cutoffs) is considered in d dimensions. It is argued that, for a certain range of τ (d/dfg < τ - 1 < d/dfs), the chains are neither Gaussian (fractal dimension df = dfg) nor fully swollen (d f = dfs), but instead obey exactly a hyperscaling relation, df = d/(τ - 1). This result applies both to linear polymers (dfg = 2, d fs = 1/ν(d)) and to « polymeric fractals » (e.g., sol-molecules) of general connectivity. Scaling laws for dilution are also given.