Geometry of Random Cayley Graphs of Abelian Groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909835575951360 |
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| author | Hermon, Jonathan Olesker-Taylor, Sam |
| author_facet | Hermon, Jonathan Olesker-Taylor, Sam |
| contents | Consider the random Cayley graph of a finite Abelian group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$. Draw a vertex $U \sim \operatorname{Unif}(G)$.
We show that the graph distance $\operatorname{dist}(\mathsf{id},U)$ from the identity to $U$ concentrates at a particular value $M$, which is the minimal radius of a ball in $\mathbb Z^k$ of cardinality at least $|G|$, under mild conditions. In other words, the distance from the identity for all but $o(|G|)$ of the elements of $G$ lies in the interval $[M - o(M), M + o(M)]$. In the regime $k \gtrsim \log |G|$, we show that the diameter of the graph is also asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis (1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure of $G$.
Write $d(G)$ for the minimal size of a generating subset of $G$. We prove that the order of the spectral gap is $|G|^{-2/k}$ when $k - d(G) \asymp k$ and $|G|$ lies in a density-$1$ subset of $\mathbb N$ or when $k - 2 d(G) \asymp k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (1994).
The aforementioned results all hold with high probability over the random Cayley graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_02801 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometry of Random Cayley Graphs of Abelian Groups Hermon, Jonathan Olesker-Taylor, Sam Probability Combinatorics 05C12, 05C48, 05C80, 60B15, 60K37 Consider the random Cayley graph of a finite Abelian group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$. Draw a vertex $U \sim \operatorname{Unif}(G)$. We show that the graph distance $\operatorname{dist}(\mathsf{id},U)$ from the identity to $U$ concentrates at a particular value $M$, which is the minimal radius of a ball in $\mathbb Z^k$ of cardinality at least $|G|$, under mild conditions. In other words, the distance from the identity for all but $o(|G|)$ of the elements of $G$ lies in the interval $[M - o(M), M + o(M)]$. In the regime $k \gtrsim \log |G|$, we show that the diameter of the graph is also asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis (1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure of $G$. Write $d(G)$ for the minimal size of a generating subset of $G$. We prove that the order of the spectral gap is $|G|^{-2/k}$ when $k - d(G) \asymp k$ and $|G|$ lies in a density-$1$ subset of $\mathbb N$ or when $k - 2 d(G) \asymp k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (1994). The aforementioned results all hold with high probability over the random Cayley graph. |
| title | Geometry of Random Cayley Graphs of Abelian Groups |
| topic | Probability Combinatorics 05C12, 05C48, 05C80, 60B15, 60K37 |
| url | https://arxiv.org/abs/2102.02801 |