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| Format: | Preprint |
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2017
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| Online-Zugang: | https://arxiv.org/abs/1701.00319 |
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| _version_ | 1866911945922183168 |
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| author | Aguirre, Ander Lyu, Hanbaek Sivakoff, David |
| author_facet | Aguirre, Ander Lyu, Hanbaek Sivakoff, David |
| contents | We investigate two discrete models of excitable media on a one-dimensional integer lattice $\mathbb{Z}$: the $κ$-color Cyclic Cellular Automaton (CCA) and the $κ$-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from $\mathbb{Z}/κ\mathbb{Z}$. Neighboring sites with colors within a specified interaction range $r$ tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on $\mathbb{Z}$ as we vary the interaction range $r$. First, if $r$ is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph $\mathbb{Z}$ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If $r$ is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range $r=\lfloor κ/2 \rfloor$, we show the density of edges of differing colors at time $t$ is $Θ(t^{-1/2})$ and each site excites $Θ(t^{1/2})$ times up to time $t$. Lastly, if $r$ is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with $r=\lfloor 2/κ\rfloor+1$, we show that every site will become $(κ+1)$-periodic eventually. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1701_00319 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Phase transition in one-dimensional excitable media with variable interaction range Aguirre, Ander Lyu, Hanbaek Sivakoff, David Probability We investigate two discrete models of excitable media on a one-dimensional integer lattice $\mathbb{Z}$: the $κ$-color Cyclic Cellular Automaton (CCA) and the $κ$-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from $\mathbb{Z}/κ\mathbb{Z}$. Neighboring sites with colors within a specified interaction range $r$ tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on $\mathbb{Z}$ as we vary the interaction range $r$. First, if $r$ is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph $\mathbb{Z}$ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If $r$ is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range $r=\lfloor κ/2 \rfloor$, we show the density of edges of differing colors at time $t$ is $Θ(t^{-1/2})$ and each site excites $Θ(t^{1/2})$ times up to time $t$. Lastly, if $r$ is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with $r=\lfloor 2/κ\rfloor+1$, we show that every site will become $(κ+1)$-periodic eventually. |
| title | Phase transition in one-dimensional excitable media with variable interaction range |
| topic | Probability |
| url | https://arxiv.org/abs/1701.00319 |