Local Central Limit Theorem for Determinantal Point Processes

Forrester, Peter J.; Lebowitz, Joel L.

doi:10.1007/s10955-014-1071-2

Cited by 17 publications

(17 citation statements)

References 36 publications

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“…1). Our method also allows us to compute, for arbitrary L, the full PDF of N L for large N, which was known to be a Gaussian but only on a scale Oð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi lnðNLÞ p Þ, in the regime L ∼ Oð1=NÞ [14][15][16][17]. Our results can be summarized as follows.…”

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confidence: 86%

Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices

et al. 2014

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We consider N × N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-√2],√2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P(N,L)(N(L)) that N(L) eigenvalues lie within the box [-L,L]. This probability scales as P(N,L)(N(L) = κ(L)N) ≈ exp(-βN(2)ψ(L)(κ(L))), where β is the Dyson index of the ensemble and ψ(L)(κ(L)) is a β-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N(-1)≪L < √2 (bulk), (ii) L∼√2 on a scale of O(N(-2/3)) (edge), and (iii) L > sqrt[2] (tail). We find a dramatic nonmonotonic behavior of the number variance V(N)(L) as a function of L: after a logarithmic growth ∝ln(NL) in the bulk (when L∼O(1/N)), V(N)(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > sqrt[2]. This "dropoff" of V(N)(L) at the edge is described by a scaling function V(β) that smoothly interpolates between the bulk (i) and the tail (iii). For β = 2 we compute V(2) explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for β = 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.

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confidence: 86%

Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices

et al. 2014

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“…The gap probability inter-relation implied by (63) is exactly (11), even though the matrix ensemble inter-relation (10) used in its previous derivation is distinct from (63). This is a concrete example of the general fact that the family of gap-probability inter-relations specified by Theorem 4 or by Theorem 6 do not contain enough information to determine a particular matrix ensemble inter-relation, even though they are suggestive.…”

Section: Circular Ensemblesmentioning

confidence: 99%

“…Another advantage is that the gap probabilities for determinantal point processes can be shown to obey a local limit theorem in an appropriate asymptotic regime. The inter-relations then allow for the deduction of such asymptotic behaviour for the sum of neighbouring gap probabilities in COE n , for which no direct methods are known [11].…”

Section: Circular Ensemblesmentioning

confidence: 99%

Singular values and evenness symmetry in random matrix theory

Bornemann

Forrester

2015

Forum Mathematicum

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Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values -whereby only even, or odd, labels are observed -for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types -Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

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“…Finally, we show that certain of the above conditions are satisfied by many graph-counting polynomials and statistical mechanical systems-for example, unbranched polymers-and hence obtain a CLT or LCLT in these cases. The result mentioned in 2(a) above has also been used [9] to establish an LCLT for determinantal point processes.…”

Section: Introductionmentioning

confidence: 99%

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

Lebowitz

Pittel

Ruelle

et al. 2016

Journal of Combinatorial Theory, Series A

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We consider the asymptotic normalcy of families of random variables X which count the number of occupied sites in some large set. We write Prob(X = m) = p m z m 0 /P (z 0 ), where P (z) is the generating function P (z) = N j=0 p j z j and z 0 > 0. We give sufficient criteria, involving the location of the zeros of P (z), for these families to satisfy a central limit theorem (CLT) and even a local CLT (LCLT); the theorems hold in the sense of estimates valid for large N (we assume that Var(X) is large when N is). For example, if all the zeros lie in the closed left half plane then X is asymptotically normal, and when the zeros satisfy some additional conditions then X satisfies an LCLT. We apply these results to cases in which X counts the number of edges in the (random) set of "occupied" edges in a graph, with constraints on the number of occupied edges attached to a given vertex. Our results also apply to systems of interacting particles, with X counting 1 the number of particles in a box Λ whose size |Λ| approaches infinity; P (z) is then the grand canonical partition function and its zeros are the Lee-Yang zeros.

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Local Central Limit Theorem for Determinantal Point Processes

Cited by 17 publications

References 36 publications

Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices

Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices

Singular values and evenness symmetry in random matrix theory

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

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