论文标题
多彩色东模型
The multicolour East model
论文作者
论文摘要
我们考虑了多色东模型，这是一种与$ \ mathbb {z}^d $相关的玻璃液体模型。状态空间$ {（g \ cup \ {\ star \}）}^{\ mathbb {z}^d} $由$ | g | \ le 2^d $不同的空位类型和中性状态$ \ star $组成。对于g $中的每个$ h \，我们关联唯一的促进机制$ {\ {c_x^{h} \}}} _ {x \ in \ Mathbb {z}^d} $，与东部模型约束的旋转版本相对应。如果满足了$ c_x^{h} $，则$ x $ can can上的状态从$ h $转换为$ p \ in（0,1）$ in（0,1）$或vice vices in（0,1）或vice $ q_h \ in（0,1）$，其中$ q_h \ q_h \ q_h \ neq q_ q_ Q_ Q_ {h'} $ if $ h'\ neq h'\ neq h $ neq h'\ neq h $。值得注意的是，州的顶点$ h $不能直接过渡到$ h'\ neq h $，而相邻的$ h'$ - 空缺不会在满足$ c_x^{h} $的情况下做出贡献。因此，不同类型的空缺之间存在一种新颖的阻塞机制。我们在模型几何形状上找到足够的条件，可以具有正频谱间隙，并证明使用$ | g | = 2^d $，模型不是ergodic。对于$ d = 2 $，我们证明具有$ | g | \ le 3 $的模型具有正频谱差距，并且我们发现，在$ \ m m mathbb {z}^2 $上，East模型的光谱差距以前顺序给出的光谱差距的过渡速率有足够的条件$ q _ {\ min} \ rightarrow 0 $。特别是，当有$ h \ in g $带有$ q_h \ gg q _ {\ min} $的$ h \ in时，我们会证明这一点，通过明确构建频繁的空位类型的机制，可以配合使用最不常见空位的东部运动。
We consider the multicolour East model, a model of glass forming liquids closely related to the East model on $\mathbb{Z}^d$. The state space ${(G\cup \{\star\})}^{\mathbb{Z}^d}$ consists of $|G|\le 2^d$ different vacancy types and the neutral state $\star$. To each $h\in G$ we associate unique facilitation mechanisms ${\{c_x^{h}\}}_{x\in \mathbb{Z}^d}$ that correspond to rotated versions of the East model constraints. If $c_x^{h}$ is satisfied, the state on $x$ can transition from $h$ to $\star$ with rate $p\in (0,1)$ or vice versa with rate $q_h\in (0,1)$, where generally $q_h\neq q_{h'}$ if $h'\neq h$. Notably, vertices in the state $h$ cannot transition directly to $h'\neq h$ and neighbouring $h'$-vacancies do not contribute in satisfying $c_x^{h}$. Thus, there is a novel blocking mechanism between vacancies of differing type. We find sufficient conditions on the model geometry to have a positive spectral gap and prove that with $|G|=2^d$ the model is not ergodic. For $d=2$ we prove that the model with $|G|\le 3$ has positive spectral gap and we find sufficient conditions on the transition rates for the spectral gap to be given in the leading order by the spectral gap of the East model on $\mathbb{Z}^2$ with parameter $q_{\min}=\min_{h\in G}q_h$ in the limit $q_{\min}\rightarrow 0$. In particular, we prove this when there are $h\in G$ with $q_h\gg q_{\min}$ by explicitly constructing mechanisms on which the frequent vacancy types cooperate to facilitate the East movement of the least frequent vacancies.
