91 research outputs found

    Strong noise sensitivity and random graphs

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    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 pcp_c. 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 pc0p_c\to0 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

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    An rr-uniform hypergraph HH is semi-algebraic of complexity t=(d,D,m)\mathbf{t}=(d,D,m) if the vertices of HH correspond to points in Rd\mathbb{R}^{d}, and the edges of HH are determined by the sign-pattern of mm degree-DD polynomials. Semi-algebraic hypergraphs of bounded complexity provide a general framework for studying geometrically defined hypergraphs. The much-studied semi-algebraic Ramsey number Rrt(s,n)R_{r}^{\mathbf{t}}(s,n) denotes the smallest NN such that every rr-uniform semi-algebraic hypergraph of complexity t\mathbf{t} on NN vertices contains either a clique of size ss, or an independent set of size nn. 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 kk with an xx on the top. This bound is also the best possible if min{d,D,m}\min\{d,D,m\} is sufficiently large with respect to rr. They conjectured that in the asymmetric case, we have R3t(s,n)<nO(1)R_{3}^{\mathbf{t}}(s,n)<n^{O(1)} for fixed ss. We refute this conjecture by showing that R3t(4,n)>n(logn)1/3o(1)R_{3}^{\mathbf{t}}(4,n)>n^{(\log n)^{1/3-o(1)}} for some complexity t\mathbf{t}. 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 D=1D=1. In particular, we prove that Rrd,1,m(n,n)2O(n4r2m2)R_{r}^{d,1,m}(n,n)\leq 2^{O(n^{4r^2m^2})}, while from below, we establish Rr1,1,1(n,n)2Ω(nr/21)R^{1,1,1}_{r}(n,n)\geq 2^{\Omega(n^{\lfloor r/2\rfloor-1})}.Comment: 23 pages, 1 figur

    The complexity of approximately counting in 2-spin systems on kk-uniform bounded-degree hypergraphs

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    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 Δ\Delta-regular tree. Our objective is to study the impact of this classification on unweighted 2-spin models on kk-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 kk-ary Boolean function ff there exists a degree bound Δ0\Delta_0 so that for all ΔΔ0\Delta \geq \Delta_0 the following problem is NP-hard: given a kk-uniform hypergraph with maximum degree at most Δ\Delta, approximate the partition function of the hypergraph 2-spin model associated with ff. It is NP-hard to approximate this partition function even within an exponential factor. By contrast, if ff is a trivial symmetric Boolean function (e.g., any function ff 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

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    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 x+yx+y which is only defined when x+y1x+y\leq 1 and an involution x1xx\mapsto 1-x 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 ω\omega-complete, i.e. where every increasing sequence has a supremum. We show that any ω\omega-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 ω\omega-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

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    Given two posets P,QP,Q we say that QQ is PP-free if QQ does not contain a copy of PP. The size of the largest PP-free family in 2[n]2^{[n]}, denoted by La(n,P)La(n,P), has been extensively studied since the 1980s. We consider several related problems. Indeed, for posets PP whose Hasse diagrams are trees and have radius at most 22, we prove that there are 2(1+o(1))La(n,P)2^{(1+o(1))La(n,P)} PP-free families in 2[n]2^{[n]}, thereby confirming a conjecture of Gerbner, Nagy, Patk\'os and Vizer [Electronic Journal of Combinatorics, 2021] in these cases. For such PP we also resolve the random version of the PP-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 2[n]2^{[n]} of size sufficiently above La(n,P)La(n,P) robustly contain PP, for any poset PP whose Hasse diagram is a tree
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