The voter model is a widely used framework in sociophysics to model opinion formation based on local interactions between individuals. In this work, we investigate how the spread of consensus is affected by introducing long-range interactions. Specifically, we study a one-dimensional voter model where a fraction γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} of links connect individuals at distances r drawn from a distribution decaying as r-σ-1\documentclass[12pt]{minimal}
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\begin{document}$$r^{-\sigma -1}$$\end{document}. Our results reveal that even a small fraction of long-range interactions fundamentally alters the system’s asymptotic behavior. When long-range interactions decay rapidly σ>2\documentclass[12pt]{minimal}
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\begin{document}$$\sigma > 2$$\end{document}, their influence is restricted to distances beyond a time-dependent threshold, r∗(t)\documentclass[12pt]{minimal}
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\begin{document}$$r^*(t)$$\end{document}. For r<r∗(t)\documentclass[12pt]{minimal}
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\begin{document}$$r < r^*(t)$$\end{document}, the system exhibits short-range dynamics characterized by a Gaussian-like correlation function and a diffusion-driven growth of the correlation length, L(t)∼t1/2\documentclass[12pt]{minimal}
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\begin{document}$$L(t) \sim t^{1/2}$$\end{document}. However, for r>r∗(t)\documentclass[12pt]{minimal}
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\begin{document}$$r > r^*(t)$$\end{document}, the correlation function transitions to a power-law decay, r-σ-1\documentclass[12pt]{minimal}
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\begin{document}$$r^{-\sigma -1}$$\end{document}, highlighting the capacity of long-range links to propagate consensus across greater distances. When long-range interactions decay more slowly (σ<2\documentclass[12pt]{minimal}
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\begin{document}$$\sigma < 2$$\end{document}), they dominate the dynamics at all scales, leading to behavior akin to a system with only long-range interactions. Notably, in the regime σ<1\documentclass[12pt]{minimal}
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\begin{document}$$\sigma < 1$$\end{document} long-range links induce a stationary steady state, even for small γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}.
