64 research outputs found
The Janson inequalities for general up-sets
Janson and Janson, Luczak and Rucinski proved several inequalities for the
lower tail of the distribution of the number of events that hold, when all the
events are up-sets (increasing events) of a special form - each event is the
intersection of some subset of a single set of independent events (i.e., a
principal up-set). We show that these inequalities in fact hold for arbitrary
up-sets, by modifying existing proofs to use only positive correlation,
avoiding the need to assume positive correlation conditioned on one of the
events.Comment: 5 pages. Added weighted varian
Logarithmic Representability of Integers as k-Sums
A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each
element in {0,1,...,kn} can be written as a k-sum of elements of A in at least
one way. Seeking multiple representations as k-sums, and given any function
phi(n), with lim(phi(n))=infinity, we say that A is a truncated
phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n]
the number of ways that j can be represented as a k-sum of elements of A_{k,n}
is Theta(phi(n)). In this paper, we follow tradition and focus on the case
phi(n)=log n, and show that a randomly selected set in an appropriate
probability space is a truncated log-representative basis with probability that
tends to one as n tends to infinity. This result is a finite version of a
result proved by Erdos (1956) and extended by Erdos and Tetali (1990).Comment: 18 page
Sharp Threshold Asymptotics for the Emergence of Additive Bases
A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if
each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each
integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus
getting a random set A, what is the probability that we have obtained a
2-additive basis? We address this question when the target sum-set is
[(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some
0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation
is used, in conjunction with Janson's inequalities, to tease out a very sharp
threshold for the emergence of a 2-additive basis. Generalizations to
k-additive bases are then given.Comment: 22 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
The t-stability number of a random graph
Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at most t.
The t-stability number of G is the maximum order of a t-stable set in G. We
investigate the typical values that this parameter takes on a random graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed
non-negative integer t, we show that, with probability tending to 1 as n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic expression for
the expected number of t-stable sets of order k. We derive this expression by
performing a precise count of the number of graphs on k vertices that have
maximum degree at most k. Using the above results, we also obtain asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the vertex set of
the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version
apart from formatting and a minor amendment to Lemma 8 (and its proof on p.
21
Moderate deviations via cumulants
The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and -statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and random point fields.Comment: 24 page
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