Homology-changing percolation transitions on finite graphs
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arXiv
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| Formato: | Preprint |
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2020
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| _version_ | 1866914838168469504 |
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| author | Woolls, Michael Pryadko, Leonid |
| author_facet | Woolls, Michael Pryadko, Leonid |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_02603 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Homology-changing percolation transitions on finite graphs Woolls, Michael Pryadko, Leonid Mathematical Physics Statistical Mechanics 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. |
| title | Homology-changing percolation transitions on finite graphs |
| topic | Mathematical Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2011.02603 |