The Deletion Method For Upper Tail Estimates

Janson, Svante; Ruciński, Andrzej

doi:10.1007/s00493-004-0038-3

Cited by 55 publications

(130 citation statements)

References 19 publications

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“…Together with the definition of c in (12), and observing that µ H = ω(1), the last two inequalities imply that Pr[G n,p / ∈ S ∩ T ] is bounded as claimed in (14). This concludes the proof of Theorem 7.…”

Section: Proof Of Theoremsupporting

confidence: 66%

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General deletion lemmas via the Harris inequality

Spöhel

Steger

Warnke

2013

Journal of Combinatorics

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Abstract. We present a new approach to the deletion method that is based on the Harris inequality. We obtain deletion lemmas that are similar in spirit to those of Rödl and Ruciński, but hold for arbitrary decreasing properties. That is, we show that under appropriate conditions, with very high ('Janson-like') probability it suffices to delete a small fraction of the edges of a random graph G n,p to ensure that a given decreasing property holds. We also obtain stronger results for decreasing properties that only depend on edges involved in copies of some given graph H. As an application of our methods, we present a new deletion lemma that concerns local subgraph counts, i.e., the number of copies of H each edge or vertex of G n,p is contained in.

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Section: Proof Of Theoremsupporting

confidence: 66%

“…in our and similar combinatorial settings [4,6,5,12,13,14]. In [12], a general exponent f (n, p, H, ε) was given that is best possible up to logarithmic factors.…”

Section: Tail Bounds For Subgraph Countsmentioning

confidence: 97%

General deletion lemmas via the Harris inequality

Spöhel

Steger

Warnke

2013

Journal of Combinatorics

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“…Originally, this approach was used in the context of partition properties of random graphs [15]. Recently, it was developed further in [10], where it is shown to often yield essentially the same results as the method of Section 2.2.…”

Section: The Deletion Methodmentioning

confidence: 82%

The infamous upper tail

Janson

́ski

2002

Random Struct Algorithms

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135

156

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Let ⌫ be a finite index set and k Ն 1 a given integer. Let further ʕ [⌫]Յk be an arbitrary family of k element subsets of ⌫. Consider a (binomial) random subset ⌫ p of ⌫, where p ϭ ( p i : i ʦ ⌫) and a random variable X counting the elements of that are contained in this random subset. In this paper we survey techniques of obtaining upper bounds on the upper tail probabilities ‫(ސ‬X Ն ϩ t) for t Ͼ 0. Seven techniques, ranging from Azuma's inequality to the purely combinatorial deletion method, are described, illustrated, and compared against each other for a couple of typical applications. As one application, we obtain essentially optimal bounds for the upper tails for the numbers of subgraphs isomorphic to K 4 or C 4 in a random graph G(n, p), for certain ranges of p.

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“…It frequently serves as a test-bed for new probabilistic estimates (see, e.g., [2,15,27,21,18,17,7]), and we shall use it to demonstrate the applicability of our bootstrapping approaches. In fact, we consider the more general random hypergraph G n is included, independently, with probability p. Given a k-uniform hypergraph H, or briefly k-graph, we define X H = X H (n, p) as the number of copies of H in G (k) n,p , where by a copy we mean, as usual, a subgraph isomorphic to H. Furthermore, we write e H = |E(H)| and v H = |V (H)| for the number of edges and vertices of H, respectively.…”

Section: Main Examplementioning

confidence: 99%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

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The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability P(X (1 − ε)EX) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ε = 1) of Janson, Luczak and Ruciński.

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The Deletion Method For Upper Tail Estimates

Cited by 55 publications

References 19 publications

General deletion lemmas via the Harris inequality

General deletion lemmas via the Harris inequality

The infamous upper tail

The Lower Tail: Poisson Approximation Revisited

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