Weighted dependency graphs and the Ising model

Dousse, Jehanne; Féray, Valentin

doi:10.4171/aihpd/78

Cited by 5 publications

(7 citation statements)

References 27 publications

(50 reference statements)

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“…where the sum runs over multiset partitions π of the multiset I 1 · · · I r such that (30) π ∨ I 1 , · · · , I r = I 1 · · · I r .…”

Section: 3mentioning

confidence: 99%

Weighted dependency graphs

Féray

2018

Electron. J. Probab.

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The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables.To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model G(n, M ), uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vallée on the asymptotic normality of subword counts in random texts generated by a Markovian source.2010 Mathematics Subject Classification. Primary: 60F05. Secondary: 60C05, 05C80, 82C05, 60J10.

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“…where the sum runs over multiset partitions π of the multiset I 1 · · · I r such that (30) π ∨ I 1 , · · · , I r = I 1 · · · I r .…”

Section: 3mentioning

confidence: 99%

Weighted dependency graphs

Féray

2018

Electron. J. Probab.

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“…Similar considerations appear with weighted dependency graphs introduced by Féray [41] and in particular uniform weighted dependency graphs [44, Definition 43]. Weighted dependency graphs have been applied, for example, to the Ising model [32], and to many other examples (not covered by statistical mechanics) [41].…”

Section: Related Techniques and Applicationsmentioning

confidence: 88%

“…The generalization to weighted dependency graphs allows for applications to the Ising model as [32] shows. Already in [90] models of statistical mechanics such as the Curie-Weiss and the Ising model are studied.…”

Section: How To Bound Cumulantsmentioning

confidence: 99%

The Method of Cumulants for the Normal Approximation

Döring¹,

Jansen²,

Schubert³

2021

Preprint

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The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type |κ j (X)| ≤ j! 1+γ /∆ j−2 , which is weaker than Cramér's condition of finite exponential moments. We give a self-contained proof of some of the "main lemmas" in a book by Saulis and Statulevičius (1989), and an accessible introduction to the Cramér-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation.

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“…Cumulant Bounds : Another use of cumulant bounds to derive CLT is in [26,30] where such bounds are crucially used in the weighted dependency graph method to prove CLTs. In [26], CLTs for local and some global statistics of the Ising model (see Section 2.1.2) are proved using bounds on cumulants and a generalisation of the dependency graph method. Again, it requires restrictions on regimes for the Ising model and variance lower bounds but it is not obvious if the methods can be adapted to other similar spin models.…”

Section: Comparison With Clts On General Spacesmentioning

confidence: 99%

“…As stated earlier, we shall only be giving a sketch of the proof here as a similar theorem or variant has appeared in all the papers using the cumulant method such as [67,68,51,5,71,11,26,8]. For more details, see the proof in [11,Section 4.4].…”

Section: Further For Any Realsmentioning

confidence: 99%

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

2018

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Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of α-mixing (for local statistics) and exponential α-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.Keywords Clustering spin models · mixing random fields · central limit theorem · Cayley graphs · fast decaying covariance · Gaussian random field · determinantal point process · exponentially quasi-local statistics · cubical complexes · nearest-neighbour graphs Mathematics Subject Classification (2000) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs · 60G60 Random fields · 60F05 Central limit theorems and other weak theorems · 60D05 Geometric probability and stochastic geometry.

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Weighted dependency graphs and the Ising model

Cited by 5 publications

References 27 publications

Weighted dependency graphs

Weighted dependency graphs

The Method of Cumulants for the Normal Approximation

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

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