论文标题
常规随机图上选民模型的不一致边缘
Discordant edges for the voter model on regular random graphs
论文作者
论文摘要
我们在带有n个顶点的常规随机图上考虑两开放的选民模型和$ d \ geq 3 $。众所周知，达成共识是按时间范围n达成的，在此时间尺度上，以一种意见的顶点的体积演变为Fisher-Wright扩散。我们对不一致的边缘数量的演变感兴趣（即，将顶点与不同意见联系起来的边缘），可以将其视为具有一种意见的顶点的周长，并且是可观察到的关键，以捕获如何达成共识。我们表明，如果最初将两个意见独立于bernoulli分布$ \ in（0，1）$中的bernoulli分布，则按时尺度1不一致的边缘的比例降低并稳定在取决于D和U的价值上，并且与D和U相关，并且与两个随机步行的时间相关，这是在两个邻近的邻近dertices of Dem of Driminite of Driminite of Driminite Tree上的相关时间。此外，我们表明，按时间尺度n不一致的边缘的比例从恒定的高原移动，并以指数式的方式收敛到零。我们的证据利用了融合随机步行的经典双重系统，并利用了Cooper等人的想法。 （2010年）建立在所谓的第一次访问时段。我们进一步引入了一种新型技术，以从中等时间尺度上合并随机行走的弱依赖性来得出浓度特性。
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.
