Matches in SemOpenAlex for { <https://semopenalex.org/work/W2976593117> ?p ?o ?g. }
Showing items 1 to 97 of
97
with 100 items per page.
- W2976593117 abstract "Given a graph $G$, a probability measure $mu$ on the subsets of the edge set of $G$ is said to be $1$-independent if events determined by edge sets that are at graph distance at least $1$ apart in $G$ are independent. Call such a probability measure a $1$-ipm on $G$, and denote by $mathbf{G}_{mu}$ the associated random spanning subgraph of $G$. Let $mathcal{M}_{1,geqslant p}(G)$ (resp. $mathcal{M}_{1,leqslant p}(G)$) denote the collection of $1$-ipms $mu$ on $G$ for which each edge is included in $mathbf{G}_{mu}$ with probability at least $p$ (resp. at most $p$). Let $mathbb{Z}^2$ denote the square integer lattice. Balister and Bollobas raised the question of determining the critical value $p_{star}=p_{1,c}(mathbb{Z}^2)$ such that for all $p>p_{star}$ and all $mu in mathcal{M}_{1,geqslant p}(mathbb{Z}^2)$, $left(mathbf{mathbb{Z}^2}right)_{mu}$ almost surely contains an infinite component. This can be thought of as asking for a $1$-independent analogue of the celebrated Harris--Kesten theorem. In this paper we investigate both this problem and connectivity problems for $1$-ipms more generally. We give two lower bounds on $p_{star}$ that significantly improve on the previous bounds. Furthermore, motivated by the Russo--Seymour--Welsh lemmas, we define a $1$-independent critical probability for long paths and determine its value for the line and ladder lattices. Finally, for finite graphs $G$ we study $f_{1,G}(p)$ (respectively $F_{1,G}(p)$), the infimum (resp. supremum) over all $muin mathcal{M}_{1,geqslant p}(G)$ (resp. all $mu in mathcal{M}_{1,leqslant p}(G)$) of the probability that $mathbf{G}_{mu}$ is connected. We determine $f_{1,G}(p)$ and $F_{1,G}(p)$ exactly when $G$ is a path, a complete graph and a cycle of length at most $5$. Many new problems arise from our work, which are discussed in the final section of the paper." @default.
- W2976593117 created "2019-10-03" @default.
- W2976593117 creator A5038904202 @default.
- W2976593117 creator A5059877314 @default.
- W2976593117 creator A5074450972 @default.
- W2976593117 date "2019-09-30" @default.
- W2976593117 modified "2023-10-17" @default.
- W2976593117 title "Long paths and connectivity in {$1$}-independent random graphs" @default.
- W2976593117 cites W1965064856 @default.
- W2976593117 cites W1970811763 @default.
- W2976593117 cites W1977183350 @default.
- W2976593117 cites W1982711333 @default.
- W2976593117 cites W1986746825 @default.
- W2976593117 cites W1987767063 @default.
- W2976593117 cites W2022436302 @default.
- W2976593117 cites W2027023078 @default.
- W2976593117 cites W2031047649 @default.
- W2976593117 cites W2031121449 @default.
- W2976593117 cites W2032718312 @default.
- W2976593117 cites W2034975570 @default.
- W2976593117 cites W2037697450 @default.
- W2976593117 cites W2069513471 @default.
- W2976593117 cites W2091172840 @default.
- W2976593117 cites W2097612342 @default.
- W2976593117 cites W2111365800 @default.
- W2976593117 cites W2115080498 @default.
- W2976593117 cites W2125637323 @default.
- W2976593117 cites W2130754058 @default.
- W2976593117 cites W2144483280 @default.
- W2976593117 cites W2146868839 @default.
- W2976593117 cites W2157869188 @default.
- W2976593117 cites W2159336341 @default.
- W2976593117 cites W2546255323 @default.
- W2976593117 cites W2962728955 @default.
- W2976593117 cites W2963461446 @default.
- W2976593117 cites W765294857 @default.
- W2976593117 hasPublicationYear "2019" @default.
- W2976593117 type Work @default.
- W2976593117 sameAs 2976593117 @default.
- W2976593117 citedByCount "0" @default.
- W2976593117 crossrefType "posted-content" @default.
- W2976593117 hasAuthorship W2976593117A5038904202 @default.
- W2976593117 hasAuthorship W2976593117A5059877314 @default.
- W2976593117 hasAuthorship W2976593117A5074450972 @default.
- W2976593117 hasConcept C114614502 @default.
- W2976593117 hasConcept C118615104 @default.
- W2976593117 hasConcept C121332964 @default.
- W2976593117 hasConcept C121864883 @default.
- W2976593117 hasConcept C134306372 @default.
- W2976593117 hasConcept C21031990 @default.
- W2976593117 hasConcept C2777620828 @default.
- W2976593117 hasConcept C2780009758 @default.
- W2976593117 hasConcept C2780897414 @default.
- W2976593117 hasConcept C33923547 @default.
- W2976593117 hasConcept C41008148 @default.
- W2976593117 hasConcept C51329190 @default.
- W2976593117 hasConcept C77088390 @default.
- W2976593117 hasConceptScore W2976593117C114614502 @default.
- W2976593117 hasConceptScore W2976593117C118615104 @default.
- W2976593117 hasConceptScore W2976593117C121332964 @default.
- W2976593117 hasConceptScore W2976593117C121864883 @default.
- W2976593117 hasConceptScore W2976593117C134306372 @default.
- W2976593117 hasConceptScore W2976593117C21031990 @default.
- W2976593117 hasConceptScore W2976593117C2777620828 @default.
- W2976593117 hasConceptScore W2976593117C2780009758 @default.
- W2976593117 hasConceptScore W2976593117C2780897414 @default.
- W2976593117 hasConceptScore W2976593117C33923547 @default.
- W2976593117 hasConceptScore W2976593117C41008148 @default.
- W2976593117 hasConceptScore W2976593117C51329190 @default.
- W2976593117 hasConceptScore W2976593117C77088390 @default.
- W2976593117 hasLocation W29765931171 @default.
- W2976593117 hasOpenAccess W2976593117 @default.
- W2976593117 hasPrimaryLocation W29765931171 @default.
- W2976593117 hasRelatedWork W1605444349 @default.
- W2976593117 hasRelatedWork W1767936884 @default.
- W2976593117 hasRelatedWork W1956452579 @default.
- W2976593117 hasRelatedWork W1991889852 @default.
- W2976593117 hasRelatedWork W1993424457 @default.
- W2976593117 hasRelatedWork W1994281890 @default.
- W2976593117 hasRelatedWork W2144467924 @default.
- W2976593117 hasRelatedWork W2185938531 @default.
- W2976593117 hasRelatedWork W2553824928 @default.
- W2976593117 hasRelatedWork W2748577230 @default.
- W2976593117 hasRelatedWork W2808270073 @default.
- W2976593117 hasRelatedWork W2899218829 @default.
- W2976593117 hasRelatedWork W2911558695 @default.
- W2976593117 hasRelatedWork W2949936740 @default.
- W2976593117 hasRelatedWork W2953075938 @default.
- W2976593117 hasRelatedWork W2963628733 @default.
- W2976593117 hasRelatedWork W2964299423 @default.
- W2976593117 hasRelatedWork W2994544165 @default.
- W2976593117 hasRelatedWork W3015052924 @default.
- W2976593117 hasRelatedWork W3106070161 @default.
- W2976593117 isParatext "false" @default.
- W2976593117 isRetracted "false" @default.
- W2976593117 magId "2976593117" @default.
- W2976593117 workType "article" @default.