4,399 research outputs found
On the counting problem in inverse Littlewood--Offord theory
Let be i.i.d. Rademacher random variables
taking values with probability each. Given an integer vector
, its concentration probability is the
quantity . The Littlewood-Offord problem asks for bounds
on under various hypotheses on , whereas
the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a
characterization of all vectors for which
is large. In this paper, we study the associated
counting problem: How many integer vectors belonging to a
specified set have large ? The motivation for our study
is that in typical applications, the inverse Littlewood-Offord theorems are
only used to obtain such counting estimates. Using a more direct approach, we
obtain significantly better bounds for this problem than those obtained using
the inverse Littlewood--Offord theorems of Tao and Vu and of Nguyen and Vu.
Moreover, we develop a framework for deriving upper bounds on the probability
of singularity of random discrete matrices that utilizes our counting result.
To illustrate the methods, we present the first `exponential-type' (i.e.,
for some positive constant ) upper bounds on the singularity
probability for the following two models: (i) adjacency matrices of dense
signed random regular digraphs, for which the previous best known bound is
due to Cook; and (ii) dense row-regular -matrices, for
which the previous best known bound is for any constant
due to Nguyen
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Singularity of random symmetric matrices -- a combinatorial approach to improved bounds
Let denote a random symmetric matrix whose upper diagonal
entries are independent and identically distributed Bernoulli random variables
(which take values and with probability each). It is widely
conjectured that is singular with probability at most . On
the other hand, the best known upper bound on the singularity probability of
, due to Vershynin (2011), is , for some unspecified small
constant . This improves on a polynomial singularity bound due to
Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the
singularity probability decays faster than any polynomial. In this paper,
improving on all previous results, we show that the probability of singularity
of is at most for all sufficiently
large . The proof utilizes and extends a novel combinatorial approach to
discrete random matrix theory, which has been recently introduced by the
authors together with Luh and Samotij.Comment: Final version incorporating referee comment
Large Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension
We review an approach which aims at studying discrete (pseudo-)manifolds in
dimension and called random tensor models. More specifically, we
insist on generalizing the two-dimensional notion of -angulations to higher
dimensions. To do so, we consider families of triangulations built out of
simplices with colored faces. Those simplices can be glued to form new building
blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can
in turn be glued together to form triangulations. The main challenge is to
classify the triangulations built from a given set of bubbles with respect to
their numbers of bubbles and simplices of codimension two. While the colored
triangulations which maximize the number of simplices of codimension two at
fixed number of simplices are series-parallel objects called melonic
triangulations, this is not always true anymore when restricting attention to
colored triangulations built from specific bubbles. This opens up the
possibility of new universality classes of colored triangulations. We present
three existing strategies to find those universality classes. The first two
strategies consist in building new bubbles from old ones for which the problem
can be solved. The third strategy is a bijection between those colored
triangulations and stuffed, edge-colored maps, which are some sort of hypermaps
whose hyperedges are replaced with edge-colored maps. We then show that the
present approach can lead to enumeration results and identification of
universality classes, by working out the example of quartic tensor models. They
feature a tree-like phase, a planar phase similar to two-dimensional quantum
gravity and a phase transition between them which is interpreted as a
proliferation of baby universes
Cokernels of random matrices satisfy the Cohen-Lenstra heuristics
Let A be an n by n random matrix with iid entries taken from the p-adic
integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A
has a universal probability distribution. In particular, the p-part of an iid
random matrix over the integers has cokernel distributed according to the
Cohen-Lenstra measure up to an exponentially small error.Comment: 21 pages; submitte
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