Homology-changing percolation transitions on finite graphs

Abstract

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.

Figures (5)

• Figure 1: (Color online) Finding the erasure pseudothreshold for square lattice toric codes. Symbols show the homological error probability (\ref{['eq:prob-E']}) evaluated numerically for different graphs labeled by the number of edges $n=2d^2$, with distances $d$ ranging from $60$ to $220$, plotted as a function of open edge probability $p$. As expected, beautiful crossing point very close to $p_E^0=1/2$ is observed. Lines are the polynomials $f_n(\xi)=A_0+A_{1n}\xi+A_{2n}\xi^2+\ldots$ of $\xi=p-p^0$ of degree 6 obtained by fitting the data collectively in the range $0.49\le p\le 0.51$. The vertical dashed line indicates the square lattice percolation threshold $p_{\rm c}=1/2$.
• Figure 2: (Color online) Top: as in Fig. \ref{['fig:PE-4-4']} but for the hyperbolic code family $\{5,5\}$. The green arrow indicates the position of the crossing point found by the fit; it is significantly below the percolation threshold for the corresponding infinite lattice (vertical red dashed line). In addition, the data for the graph with $n=15\,350$ is shifted upward, which we associate with a slightly smaller ratio $d/\ln n$, see Fig. \ref{['fig:dist']}. This is verified in the bottom plot, where an additional vertical shift proportional to $\ln n/d$ is added, which substantially improves the convergence at the crossing point.
• Figure 3: Homological distance $d\equiv d_Z$ associated with non-trivial cycles for optimal graphs in $\{7,3\}$ and $\{5,5\}$ families vs. the graph size $n$ (number of edges) with the logarithmic scale. Numbers also indicate the graph sizes. Smaller relative distances $d/\ln n$ result in larger erasure probabilities in Figs. \ref{['fig:pE-hyp55']} and \ref{['fig:pE-hyp73']} (top); this can be compensated to some extend by using the correction term as in bottom plots in Figs. \ref{['fig:pE-hyp55']} and \ref{['fig:pE-hyp73']}.
• Figure 4: (Color online) As in Fig. \ref{['fig:pE-hyp55']} but for the hyperbolic code family $\{7,3\}$. The convergence at the crossing point is much better than that in Fig. \ref{['fig:pE-hyp55']} (top), which we associate with substantially higher ratios $d/\ln n$ for the graphs in this family. Bottom plot: addition of the additional vertical shift $B\ln n/d$ causes a substantial shift of the crossing point position without visibly improving the convergence.
• Figure 5: (Color online) Using the random-graph-like scaling for locating the percolation transition for hyperbolic graphs in the $\{5,5\}$ (top) and $\{7,3\}$ (middle) families, and for degree-$5$ random graphs (bottom). Values of the open bond probability $p$ where the expected size of the largest cluster equals $\omega n^{2/3}$ are plotted as a function of $n^{-1/3}$, for values of $\omega$ as indicated. Here $n$ is the number of edges in the graph. The lines intersect close to the percolation transition point, as indicated by horizontal dashed lines.

Theorems & Definitions (34)

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• Example \oldthetheorem: Anisotropic square-lattice toric codes
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