719 research outputs found
On the lower tail variational problem for random graphs
We study the lower tail large deviation problem for subgraph counts in a
random graph. Let denote the number of copies of in an
Erd\H{o}s-R\'enyi random graph . We are interested in
estimating the lower tail probability for fixed .
Thanks to the results of Chatterjee, Dembo, and Varadhan, this large
deviation problem has been reduced to a natural variational problem over
graphons, at least for (and conjecturally for a larger
range of ). We study this variational problem and provide a partial
characterization of the so-called "replica symmetric" phase. Informally, our
main result says that for every , and for some
, as slowly, the main contribution to the lower tail
probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted
edge density. On the other hand, this is false for non-bipartite and
close to 1.Comment: 15 pages, 5 figures, 1 tabl
The Bipartite Swapping Trick on Graph Homomorphisms
We provide an upper bound to the number of graph homomorphisms from to
, where is a fixed graph with certain properties, and varies over
all -vertex, -regular graphs. This result generalizes a recently resolved
conjecture of Alon and Kahn on the number of independent sets. We build on the
work of Galvin and Tetali, who studied the number of graph homomorphisms from
to when is bipartite. We also apply our techniques to graph
colorings and stable set polytopes.Comment: 22 pages. To appear in SIAM J. Discrete Mat
Sphere packing bounds via spherical codes
The sphere packing problem asks for the greatest density of a packing of
congruent balls in Euclidean space. The current best upper bound in all
sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We
revisit their argument and improve their bound by a constant factor using a
simple geometric argument, and we extend the argument to packings in hyperbolic
space, for which it gives an exponential improvement over the previously known
bounds. Additionally, we show that the Cohn-Elkies linear programming bound is
always at least as strong as the Kabatiansky-Levenshtein bound; this result is
analogous to Rodemich's theorem in coding theory. Finally, we develop
hyperbolic linear programming bounds and prove the analogue of Rodemich's
theorem there as well.Comment: 30 pages, 2 figure
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