Matches in SemOpenAlex for { <https://semopenalex.org/work/W2122060711> ?p ?o ?g. }
- W2122060711 abstract "We study a random system of $cn$ linear equations over $n$ variables in GF(2), where each equation contains exactly $r$ variables; this is equivalent to $r$-XORSAT. cite{ikkm,amxor} determined the clustering threshold, $c^*_r$: if $c=c^*_r+e$ for any constant $e>0$, then aas the solutions partition into well-connected, well-separated {em clusters} (with probability tending to 1 as $nrightarrowinfty$). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly non-rigorous techniques from statistical physics. We extend that study to the range $c=c^*_r+o(1)$, showing that if $c=c^*_r+n^{-d}, d>0$, then the connectivity parameter of each $r$-XORSAT cluster is $n^{Theta(d)}$, as compared to $O(log n)$ when $c=c^*_r+e$. This means that one can move between any two solutions in the same cluster via a sequence of solutions where consecutive solutions differ on at most $n^{Theta(d)}$ variables; this is tight up to the implicit constant. In contrast, moving to a solution in another cluster requires that some pair of consecutive solutions differ in at least $n^{1-O(d)}$ variables. Along the way, we prove that in a random $r$-uniform hypergraph with edge-density $n^{-d}$ above the $k$-core threshold, aas every vertex not in the $k$-core can be removed by a sequence of $n^{Theta(d)}$ vertex-deletions in which the deleted vertex has degree less than $k$; again, this is tight up to the implicit constant." @default.
- W2122060711 created "2016-06-24" @default.
- W2122060711 creator A5072889645 @default.
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- W2122060711 date "2013-09-19" @default.
- W2122060711 modified "2023-09-25" @default.
- W2122060711 title "Inside the clustering threshold for random linear equations" @default.
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- W2122060711 doi "https://doi.org/10.48550/arxiv.1309.6651" @default.
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