Abstract
We are concerned with how the implementation of growth determines the expected number of state-changes in a growing self-organizing process. With this problem in mind, we examine two versions of the voter model on a one-dimensional growing lattice. Our main result asserts that the expected number of state-changes before an absorbing state is found can be controlled by balancing the conservative and disruptive forces of growth. This is because conservative growth preserves the self-organization of the voter model as it searches for an absorbing state, whereas disruptive growth undermines this self-organization. In particular, we focus on controlling the expected number of state-changes as the rate of growth tends to zero or infinity in the limit. These results illustrate how growth can affect the costs of self-organization and so are pertinent to the physics of growing active matter.
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Notes
As will become apparent we could have fixed our growth rate, \(P_{g}\), and manipulated \(P_{d}\) instead. The meaningful parameter is in fact the ratio of the total decision rate and the total growth rate.
Also referred to as a nonequilibrium steady-state or fixation.
The ‘most’ disruptive form of growth is not well-defined here.
\(\beta _{c}\) can be computed efficiently by using Eq. (25) in tandem with the observation that in the limit \(P_{g} \rightarrow 0\), following any growth event \({\mathcal {C}}\) can only be in one of 2N states. From these 2N states \({\mathcal {C}}\) can only reach \(N(N+1)\) states, including the absorbing states. Therefore, K can be calculated using a matrix of \(O(N^{2})\) instead of \(O(2^{N})\). Similar reasoning applies to computing \(\beta _{c}\) for \({\mathcal {A}}\).
We now refer to the original \({\varvec{g}}\) studied as \({\varvec{g}}^{1}\).
An argument of this nature holds for any functional form of the growth greater than linear, so that \(\epsilon < O(1)\).
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Communicated by Sidney Redner.
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Ross, R.J.H., Fontana, W. Balancing Conservative and Disruptive Growth in the Voter Model. J Stat Phys 183, 15 (2021). https://doi.org/10.1007/s10955-021-02749-7
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DOI: https://doi.org/10.1007/s10955-021-02749-7