# Phase transition in random walks with long-range correlations Academic Article

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### abstract

• 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 $({D}_{L}\ensuremath{\sim}{L}^{\ensuremath{\alpha}},\ensuremath{\alpha}>1)$, 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.

• July 1, 2004