Phase transition in random walks with long-range correlations

Motivated by recent results in the theory of correlated sequences, we analyze the dynamics of random walks with long-term memory (binary chains with long-range correlations). In our model, the probability for a unit bit in a binary string depends on the fraction of unities preceding it. We show that the system undergoes a dynamical phase transition from normal diffusion, in which the variance D(L) scales as the string's length L, into a superdiffusion phase ( D(L) approximately Lalpha,alpha>1), when the correlation strength exceeds a critical value. We demonstrate the generality of our results with respect to alternative models, and discuss their applicability to various data, such as coarse-grained DNA sequences, written texts, and financial data.