A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

Suen, W. C. Stephen

doi:10.1002/rsa.3240010210

Cited by 47 publications

(39 citation statements)

References 7 publications

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“…In fact, an inspection of the proofs reveals that Theorem 1 and 2 (as well as (3), Theorem 6 and Lemma 7) remain valid for the more general correlation conditions (and setup) stated by Riordan and Warnke [23]. It would be interesting to know whether similar results also hold under the weaker dependency assumptions of Suen's inequality [28,14].…”

Section: Resultsmentioning

confidence: 80%

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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Section: Resultsmentioning

confidence: 80%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

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“…The Lovász local lemma can be formulated more generally for families of events with a dependency digraph: each event A x is independent from the σ-algebra σ(A y : y ∈ X \Γ * 3. There are other probabilistic inequalities that are expressed in terms of a dependency graph (see for instance Suen [109] or Janson [68]); it would be interesting to know if any of these have counterparts in statistical mechanics. One obstacle here is the need for a counterpart of Theorem 4.1.…”

Section: The Lovász Local Lemma 41 Hard-core Versionmentioning

confidence: 99%

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma -which provides a sufficient condition for this to occur -corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37,38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

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“…Of course, our method has the drawback that it applies to the upper tail only, but this is not serious, since bounds for the lower tail easily are obtained by other well-known methods, see Janson [1], Suen [12] and Janson [2], or the survey in [3,Chapter 2]. (See also the preprint version of the present paper [6] for a new version of Suen's inequality that applies in the setting of our basic theorem.)…”

Section: Sjmentioning

confidence: 99%

The Deletion Method For Upper Tail Estimates

2004

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Abstract. We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results.Some applications are given to the number X G of copies of a graph G in the random graph G(n, p). In particular, for G = K 4 and G = C 4 we improve the earlier known upper bounds on − ln P(X K 4 ≥ 2 E X K 4 ) in some range of p = p(n). .

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A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

Cited by 47 publications

References 7 publications

The Lower Tail: Poisson Approximation Revisited

The Lower Tail: Poisson Approximation Revisited

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

The Deletion Method For Upper Tail Estimates

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