论文标题
圆环上的Q-VOTER模型
The q-voter model on the torus
论文作者
论文摘要
在$ q $ - 投票模型中，选民以$ x $的价格以$ f_x^q $更改其意见，其中$ f_x $是邻居的一部分，有相反的意见。平均场计算表明，如果$ Q <1 $和聚类，则应在$ Q> 1 $之间存在共存。该模型已经由物理学家进行了广泛的研究，但我们不知道任何严格的结果。在本文中，我们使用选民模型扰动的机械表明，猜想的行为以$ Q $接近1的价格成立。更确切地说，如果$ q <1 $，那么对于任何$ m <\ iffty $，$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n^m $ a $ n^m $的过程始终是一个最初的固定阶段，并且始终是1/2 senter，d dementh是d d oss n d d d oss d d d oss d d d d oss d d d d d d d oss d d d.如果$ q> 1 $，则该过程迅速达到固定意见。有趣的是，在第二种情况下，限制颂歌（在其加速时间尺度上）在时间$ \ log n $达到0，但同一时间量表上的随机过程在时间$ $（1/3）\ log n $中消失。
In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q<1$ and clustering if $q>1$. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for $q$ close to 1. More precisely, we show that if $q<1$, then for any $m<\infty$ the process on the three-dimensional torus with $n$ points survives for time $n^m$, and after an initial transient phase has a density that it is always close to 1/2. If $q>1$, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time $\log n$ but the stochastic process on the same time scale dies out at time $(1/3)\log n$.
