Generalisation of the Sinai anomalous diffusion law

Sinai has considered a novel one-dimensional walk with a random bias field E on each site. He has shown that when the field is taken with equal probability to be +E₀ or -E₀ the RMS displacement R identical to (( chi ²))^1/2 increases with time t by the anomalously slow law R approximately (log t)². Here the authors introduce long-range correlation between the random fields on each site by choosing a 'string' of k sites to a have the same value of E, where k is chosen from the power law distribution P(k)=k^{- beta}. They find that the Sinai law is generalised to the form R approximately (log t)^y, where y sticks at the Sinai value y=2 for beta >2. However, for 1< beta <2, y varies continuously with beta as t= beta /( beta -1). The authors interpret this result physically in terms of the novel crossover between the physics underlying the Sinai effect and the physics of biased diffusion.