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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.13127 |
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| _version_ | 1866915069116284928 |
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| author | Peled, Yuval Peleg, Niv |
| author_facet | Peled, Yuval Peleg, Niv |
| contents | We study the maximum dimension $d=d(n,p)$ for which an Erdős-Rényi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point where the graph's minimum degree exceeds half its average degree. We show that if $p < (1-\varepsilon)p_c $, then $d(n,p)$ is asymptotically almost surely (a.a.s.) equal to the minimum degree of $G(n,p)$. In contrast, if $p_c \leq p = o(n^{-1/2}) $ then $d(n,p) $ is a.a.s. equal to $(1/2 + o(1))np$. The second result confirms, in this regime, a conjecture of Krivelevich, Lew, and Michaeli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Rigidity of Random Graphs in high-dimensional spaces Peled, Yuval Peleg, Niv Combinatorics We study the maximum dimension $d=d(n,p)$ for which an Erdős-Rényi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point where the graph's minimum degree exceeds half its average degree. We show that if $p < (1-\varepsilon)p_c $, then $d(n,p)$ is asymptotically almost surely (a.a.s.) equal to the minimum degree of $G(n,p)$. In contrast, if $p_c \leq p = o(n^{-1/2}) $ then $d(n,p) $ is a.a.s. equal to $(1/2 + o(1))np$. The second result confirms, in this regime, a conjecture of Krivelevich, Lew, and Michaeli. |
| title | On the Rigidity of Random Graphs in high-dimensional spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.13127 |