Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Lebowitz, Joel L.; Pittel, Boris; Ruelle, David; Speer, Eugène R.

doi:10.1016/j.jcta.2016.02.009

Cited by 39 publications

(34 citation statements)

References 38 publications

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“…As a consequence, they gave a simple criterion for {X * n } to be asymptotically normal. Later, Lebowitz, Pittel, Ruelle and Speer (Henceforth LPRS) [20] studied a looser restriction on the roots that guarantees a central limit theorem for X n with large variance. They showed that if σ n n −1/3 → ∞, and none of the roots of {P n (z)} is contained in a neighbourhood of 1 ∈ C, then X n satisfies a central limit theorem.…”

Section: Resultsmentioning

confidence: 99%

Central limit theorems from the roots of probability generating functions

Michelen

Sahasrabudhe

2019

Advances in Mathematics

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For each n, let Xn ∈ {0, . . . , n} be a random variable with mean µn, standard deviation σn, and let Pn(z) = n k=0 P(Xn = k)z k , be its probability generating function. We show that if none of the complex zeros of the polynomials {Pn(z)} is contained in a neighbourhood of 1 ∈ C and σn > n ε for some ε > 0, then X * n = (Xn − µn)σ −1 n is asymptotically normal as n → ∞: that is tends in distribution to a random variable Z ∼ N (0, 1).Moreover, we show this result is sharp in the sense that there exist sequences of random variables {Xn} with σn > C log n for which Pn(z) has no roots near 1 and X * n is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of Pn(z) and the distribution of the random variable Xn. 1 arXiv:1804.07696v2 [math.PR] 12 Jun 2018 P X (z) are non-negative. Hence, we may write P X (z) = n i=1 (q i z + 1 − q i ), for some q 1 , . . . q n ∈ [0, 1]. A little further thought reveals that this special expression for the probability generating function corresponds to an expression of X as a sum of independent random variables X = X 1 + · · · + X n , where X i is the {0, 1}-random variable, taking 1 with probability q i . Thus, with an appropriate central limit theorem at hand, we see that X must be approximately normal, provided the variance of X is sufficiently large. In other words, from this one piece of information (albeit a strong piece of information) about the zeros of P X (z), one can quickly deduce quite a bit of information about the distribution of X. The aim of this paper is to show that this assumption of real rootedness of P X (z) can be related quite considerably while yielding similar results. In particular, we give three different results each of which says that "if X has large variance and the roots of P X (z) avoid a region in the complex plane, then X is approximately normal." HistoryBefore turning to our contributions, we take a brief moment to situate our results in an old and well-studied field centred around the following question: What does information about the coefficients of P (z) tell us about the distribution of the complex roots of P (and vice versa)? This question has a long and rich history, reaching back to the seminal work of Littlewood, Szegő, Pólya, and perhaps even Cauchy, due to his 1829 proof [8] of the fundamental theorem of algebra, which gives explicit bounds on the magnitude of the complex roots (see [7, Theorem 1.2.1] for a modern treatment of this proof). One line of research, initiated by the 1938 -1943 work of Littlewood and Offord [21, 22, 23], concerns the typical distribution of roots of random polynomials. For example Kac [18] gave an exact integral formulafor the number of real roots of random polynomial, with coefficients sampled independently from a normal distribution. Later, Erdős and Offord [11] showed that as n → ∞ almost all polynomials of the form n i=0 ε i x i , where ε 1 , . . . , ε n ∈ {0...

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Section: Resultsmentioning

confidence: 99%

Central limit theorems from the roots of probability generating functions

Michelen

Sahasrabudhe

2019

Advances in Mathematics

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“…This condition is present in Lemma 6.4 from [LPRS16]. Already there, we do not know whether the condition is necessary.…”

Section: Resultsmentioning

confidence: 98%

“…From the definition of stability it may be shown that the generating polynomial for aX n + bY n has no zeros near 1 (this is Lemma 6.1 below). A result of [LPRS16] then implies a Gaussian approximation for aX n +bY n (Lemma 6.4 below). Tightness and continuity could be used to extend this to positive real (a, b), however the usual Cramér-Wold argument requires this for all real (a, b) regardless of sign.…”

Section: Resultsmentioning

confidence: 98%

Multivariate CLT follows from strong Rayleigh property

Ghosh¹,

Liggett²,

Pemantle³

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Let (X1, . . . , X d ) be a random nonnegative integer vector. Many conditions are known to imply a central limit theorem for a sequence of such random vectors, for example, independence and convergence of the normalized covariances, or various combinatorial conditions allowing the application of Stein's method, couplings, etc. Here, we prove a central limit theorem directly from hypotheses on the probability generating function f (z1, . . . , z d ). In particular, we show that the f being real stable (meaning no zeros with all coordinates in the open upper half plane) is enough to imply a CLT under a nondegeneracy condition on the variance. Known classes of distributions with real stable generating polynomials include spanning tree measures, conditioned Bernoullis and counts for determinantal point processes. Soshnikov [Sos02] showed that occupation counts of disjoint sets by a determinantal point process satisfy a multivariate CLT. Our results extend Soshnikov's to the class of real stable laws. The class of real stable laws is much larger than the class of determinantal laws, being defined by inequalities rather than identities. Along the way we investigate the related problem of stable multiplication.

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“…As the 2 × 2 matrix integral operator J −1 A is not self adjoint, we have no immediate information as to the location of the zeros of the generating function. We note however that whenever the zeros of Ξ(z; J) come in complex conjugate paris whose real parts are non-positive, and σ J → ∞ then the process satisfies a CLT [23].…”

Section: Goe and Gsementioning

confidence: 91%

Local Central Limit Theorem for Determinantal Point Processes

Forrester

Lebowitz

2014

J Stat Phys

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We prove a local central limit theorem (LCLT) for the number of points N (J) in a region J in R d specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N (J) tends to infinity as |J| → ∞. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for {E(k; J)} -the probabilities of there being exactly k points in Jall lie on the negative real z-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the k-th largest eigenvalue at the soft edge, and of the spacing between k-th neigbors in the bulk.

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Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Cited by 39 publications

References 38 publications

Central limit theorems from the roots of probability generating functions

Central limit theorems from the roots of probability generating functions

Multivariate CLT follows from strong Rayleigh property

Local Central Limit Theorem for Determinantal Point Processes

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