91 research outputs found
Strong noise sensitivity and random graphs
The noise sensitivity of a Boolean function describes its likelihood to flip
under small perturbations of its input. Introduced in the seminal work of
Benjamini, Kalai and Schramm [Inst. Hautes \'{E}tudes Sci. Publ. Math. 90
(1999) 5-43], it was there shown to be governed by the first level of Fourier
coefficients in the central case of monotone functions at a constant critical
probability . Here we study noise sensitivity and a natural stronger
version of it, addressing the effect of noise given a specific witness in the
original input. Our main context is the Erd\H{o}s-R\'{e}nyi random graph, where
already the property of containing a given graph is sufficiently rich to
separate these notions. In particular, our analysis implies (strong) noise
sensitivity in settings where the BKS criterion involving the first Fourier
level does not apply, for example, when polynomially fast in the
number of variables.Comment: Published at http://dx.doi.org/10.1214/14-AOP959 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Semi-algebraic and semi-linear Ramsey numbers
An -uniform hypergraph is semi-algebraic of complexity
if the vertices of correspond to points in
, and the edges of are determined by the sign-pattern of
degree- polynomials. Semi-algebraic hypergraphs of bounded complexity
provide a general framework for studying geometrically defined hypergraphs.
The much-studied semi-algebraic Ramsey number
denotes the smallest such that every -uniform semi-algebraic hypergraph
of complexity on vertices contains either a clique of size
, or an independent set of size . Conlon, Fox, Pach, Sudakov, and Suk
proved that R_{r}^{\mathbf{t}}(n,n)<\mbox{tw}_{r-1}(n^{O(1)}), where
\mbox{tw}_{k}(x) is a tower of 2's of height with an on the top. This
bound is also the best possible if is sufficiently large with
respect to . They conjectured that in the asymmetric case, we have
for fixed . We refute this conjecture by
showing that for some
complexity .
In addition, motivated by results of Bukh-Matou\v{s}ek and
Basit-Chernikov-Starchenko-Tao-Tran, we study the complexity of the Ramsey
problem when the defining polynomials are linear, that is, when . In
particular, we prove that , while
from below, we establish .Comment: 23 pages, 1 figur
The complexity of approximately counting in 2-spin systems on -uniform bounded-degree hypergraphs
One of the most important recent developments in the complexity of
approximate counting is the classification of the complexity of approximating
the partition functions of antiferromagnetic 2-spin systems on bounded-degree
graphs. This classification is based on a beautiful connection to the so-called
uniqueness phase transition from statistical physics on the infinite
-regular tree. Our objective is to study the impact of this
classification on unweighted 2-spin models on -uniform hypergraphs. As has
already been indicated by Yin and Zhao, the connection between the uniqueness
phase transition and the complexity of approximate counting breaks down in the
hypergraph setting. Nevertheless, we show that for every non-trivial symmetric
-ary Boolean function there exists a degree bound so that for
all the following problem is NP-hard: given a
-uniform hypergraph with maximum degree at most , approximate the
partition function of the hypergraph 2-spin model associated with . It is
NP-hard to approximate this partition function even within an exponential
factor. By contrast, if is a trivial symmetric Boolean function (e.g., any
function that is excluded from our result), then the partition function of
the corresponding hypergraph 2-spin model can be computed exactly in polynomial
time
A characterisation of ordered abstract probabilities
In computer science, especially when dealing with quantum computing or other
non-standard models of computation, basic notions in probability theory like "a
predicate" vary wildly. There seems to be one constant: the only useful example
of an algebra of probabilities is the real unit interval. In this paper we try
to explain this phenomenon. We will show that the structure of the real unit
interval naturally arises from a few reasonable assumptions. We do this by
studying effect monoids, an abstraction of the algebraic structure of the real
unit interval: it has an addition which is only defined when
and an involution which make it an effect algebra, in
combination with an associative (possibly non-commutative) multiplication.
Examples include the unit intervals of ordered rings and Boolean algebras.
We present a structure theory for effect monoids that are -complete,
i.e. where every increasing sequence has a supremum. We show that any
-complete effect monoid embeds into the direct sum of a Boolean algebra
and the unit interval of a commutative unital C-algebra. This gives us from
first principles a dichotomy between sharp logic, represented by the Boolean
algebra part of the effect monoid, and probabilistic logic, represented by the
commutative C-algebra. Some consequences of this characterisation are that
the multiplication must always be commutative, and that the unique
-complete effect monoid without zero divisors and more than 2 elements
must be the real unit interval. Our results give an algebraic characterisation
and motivation for why any physical or logical theory would represent
probabilities by real numbers.Comment: 12 pages. V2: Minor change
On some extremal and probabilistic questions for tree posets
Given two posets we say that is -free if does not contain a
copy of . The size of the largest -free family in , denoted by
, has been extensively studied since the 1980s. We consider several
related problems. Indeed, for posets whose Hasse diagrams are trees and
have radius at most , we prove that there are -free
families in , thereby confirming a conjecture of Gerbner, Nagy,
Patk\'os and Vizer [Electronic Journal of Combinatorics, 2021] in these cases.
For such we also resolve the random version of the -free problem, thus
generalising the random version of Sperner's theorem due to Balogh, Mycroft and
Treglown [Journal of Combinatorial Theory Series A, 2014], and Collares Neto
and Morris [Random Structures and Algorithms, 2016]. Additionally, we make a
general conjecture that, roughly speaking, asserts that subfamilies of
of size sufficiently above robustly contain , for any
poset whose Hasse diagram is a tree
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