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Space Representation of Stochastic Processes with Delay

Silvio R. Dahmen·Haye Hinrichsen·Wolfgang Kinzel
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Abstract

We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.

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Silvio R. Dahmen, Haye Hinrichsen, Wolfgang Kinzel. 2007-03-22. Space Representation of Stochastic Processes with Delay. https://doi.org/10.1103/physreve.77.031106

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