Phase-Transition in Binary Sequences with Long-Range Correlations
Shahar Hod, Uri Keshet
Abstract
Motivated by novel 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 DL scales as the string's length L, into a super-diffusion phase (DL ~ L1+|alpha|), 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.
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