35 research outputs found
Upper tails for triangles
With the number of triangles in the usual (Erd\H{o}s-R\'enyi) random
graph , and , we show (for some )
\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].
This is tight up to the value of .Comment: 10 page
The missing log in large deviations for triangle counts
This paper solves the problem of sharp large deviation estimates for the
upper tail of the number of triangles in an Erdos-Renyi random graph, by
establishing a logarithmic factor in the exponent that was missing till now. It
is possible that the method of proof may extend to general subgraph counts.Comment: 15 pages. Title changed. To appear in Random Structures Algorithm
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
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of