正则随机图上投票者模型的不一致边
Discordant edges for the voter model on regular random graphs
AI总结：
本文研究正则随机图上双意见投票者模型的不一致边，证明其在时间尺度1稳定到与树上随机游走相遇时间相关的值，并在时间尺度n指数衰减至零。
摘要：
我们考虑具有 n 个顶点且度为 d ≥ 3 的正则随机图上的双意见投票者模型。已知共识在时间尺度 n 上达成，并且在该时间尺度上，持有某一种意见的顶点集合的体积会演化为一个 Fisher-Wright 扩散。我们关注不一致边（即连接持有不同意见的顶点的边）数量的演化；它可以被视为持有某一种意见的顶点集合的周长，也是捕捉共识如何达成的关键可观测量。我们证明，如果初始时两种意见独立地从参数为 u ∈ (0,1) 的 Bernoulli 分布中抽取，那么在时间尺度 1 上，不一致边的比例会下降并稳定到一个依赖于 d 和 u 的值，并且该值与度为 d 的无限树上两条从相邻顶点出发的随机游走的相遇时间有关。此外，我们证明，在时间尺度 n 上，不一致边的比例会离开这一常数平台，并以指数方式收敛到零。我们的证明利用了经典的合并随机游走对偶系统，并采用了 Cooper 等人（2010）基于所谓 First Visit Time Lemma（首次访问时间引理）的思想。我们还引入了一种新技术，用于从中等时间尺度上合并随机游走的弱相依性导出集中性质。
英文摘要：
We consider the two-opinion voter model on a regular random graph with n vertices and degree $d \geq 3$. It is known that consensus is reached on time scale n and that on this time scale the volume of the set of vertices with one opinion evolves as a Fisher-Wright diffusion. We are interested in the evolution of the number of discordant edges (i.e., edges linking vertices with different opinions), which can be thought as the perimeter of the set of vertices with one opinion, and is the key observable capturing how consensus is reached. We show that if initially the two opinions are drawn independently from a Bernoulli distribution with parameter $u \in (0, 1)$, then on time scale 1 the fraction of discordant edges decreases and stabilises to a value that depends on d and u, and is related to the meeting time of two random walks on an infinite tree of degree d starting from two neighbouring vertices. Moreover, we show that on time scale n the fraction of discordant edges moves away from the constant plateau and converges to zero in an exponential fashion. Our proofs exploit the classical dual system of coalescing random walks and use ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma. We further introduce a novel technique to derive concentration properties from weak-dependence of coalescing random walks on moderate time scales.
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