论文标题
有限图上的同源性渗透过渡
Homology-changing percolation transitions on finite graphs
论文作者
论文摘要
我们考虑在序列$（\ MATHCAL {G} _t）_t _t $的有限图上的同源边缘渗透，该图由无限（Quasi）及其透射图$ \ MATHCAL {h} $覆盖，并且微弱地收敛到$ \ Mathcal {H} $。也就是说，我们使用覆盖地图将图表上的$ 1 $ -CYCLE分类为$ \ Mathcal {G} _T $作为同种琐事或非平凡的，并定义了与公开子学上如此定义的第一个同源组相关的几个阈值。我们确定同源距离$ d_t $的增长，这是$ \ nathcal {g} _t $的非平凡周期的最小尺寸，是确定同源性变化阈值位置的主要因素。特别是，我们表明，巨大的循环擦除阈值$ p_e^0 $（与广义折磨代码相应顺序的常规擦除阈值有关）与边缘渗透阈值$ p _ {\ rm c}（\ rm c}（\ rm c}） of edges of $\mathcal{G}_t$, and we give evidence that $p_E^0<p_{\rm c}(\mathcal{H})$ in several cases where this ratio remains bounded, which is necessarily the case if $\mathcal{H}$ is non-amenable.
We consider homological edge percolation on a sequence $(\mathcal{G}_t)_t$ of finite graphs covered by an infinite (quasi)transitive graph $\mathcal{H}$, and weakly convergent to $\mathcal{H}$. Namely, we use the covering maps to classify $1$-cycles on graphs $\mathcal{G}_t$ as homologically trivial or non-trivial, and define several thresholds associated with the rank of thus defined first homology group on the open subgraphs. We identify the growth of the homological distance $d_t$, the smallest size of a non-trivial cycle on $\mathcal{G}_t$, as the main factor determining the location of homology-changing thresholds. In particular, we show that the giant cycle erasure threshold $p_E^0$ (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold $p_{\rm c}(\mathcal{H})$ if the ratio $d_t/\ln n_t$ diverges, where $n_t$ is the number of edges of $\mathcal{G}_t$, and we give evidence that $p_E^0<p_{\rm c}(\mathcal{H})$ in several cases where this ratio remains bounded, which is necessarily the case if $\mathcal{H}$ is non-amenable.
