Multicritical microscopic spectral correlators of hermitian and complex matrices

Akemann, Gernot; Damgaard, Poul H.; Magnea, U.; Nishigaki, Shinsuke M.

doi:10.1016/s0550-3213(98)00143-6

Cited by 61 publications

(117 citation statements)

References 49 publications

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“…One important common ingredient is that eigenvalue correlations appear to be insensitive to the details of the underlying Hamiltonian. The success of RMT is based on this type of universality, and it is no surprise that it has received a great deal of attention in recent literature [3][4][5][6][7][8][9][10] [ [11][12][13][14][15][16][17][18][19][20] [21][22][23][24][25][26][27]. What has been shown is that spectral correlators on the scale of the average eigenvalue spacing are insensitive to the details of the probability distribution of the matrix elements.…”

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confidence: 99%

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Sener

Verbaarschot

1998

Phys. Rev. Lett.

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In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real (β = 1) and quaternion real (β = 4) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles (β = 2). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Brézin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for β = 2 as shown by Akemann, Damgaard, Magnea and Nishigaki.PACS numbers: 64.60. Cn, 05.45.+b, 11.30.Rd, 12.38.Aw Since its introduction in nuclear physics [1], Random Matrix Theory (RMT) has been applied successfully to many different branches of physics ranging from atomic physics to quantum gravity (for a recent comprehensive review we refer to [2]). One important common ingredient is that eigenvalue correlations appear to be insensitive to the details of the underlying Hamiltonian. The success of RMT is based on this type of universality, and it is no surprise that it has received a great deal of attention in recent literature [3-10] [11-20] [21-27]. What has been shown is that spectral correlators on the scale of the average eigenvalue spacing are insensitive to the details of the probability distribution of the matrix elements. Because of its mathematical simplicity most studies were performed for complex (β = 2) Hermitean RMT's. However, in the case of the classical RMT's it was shown that universality extends to real (β = 1) and quaternion real (β = 4) matrix ensembles [4,6,7]. This suggests that relations between correlation functions for different values of β which can be derived for a Gaussian probability distribution [28,29] might be valid for a wide class of probability distributions. The main goal of this paper is to establish such general relations. As a consequence, universality for the much simpler complex ensembles implies universality for the real and quaternion real ensembles.In this letter we address the question of microscopic universality for the chiral ensembles. These ensembles are relevant for the description of spectral correlations of the QCD Dirac operator. They also appear in theory of universal conductance fluctuations in mesoscopic systems [30,31]. In particular, they can be applied to spectral correlations near λ = 0. According to the Banks-Casher formula [32], this part of the spectrum is directly related to the order parameter Σ of the chiral phase transition (Σ = lim πρ(0)/V , where V is the volume of space time and ρ(λ) = k δ(λ−λ k )). It is therefore natural to introduce the microscopic limit where the variable u = λV Σ is kept fixed for V → ∞. For example, the microscopic spectral density is defined by [33] ρ S (u) = limwhere the average is over the distribution of the matrix elements of the D...

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confidence: 99%

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Sener

Verbaarschot

1998

Phys. Rev. Lett.

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“…In order to make firmer the correspondence between the matrix model and the two-dimensional IIA superstrings, it is important to proceed computing correlation functions among various matrix-model operators at higher genera and compare the results with the corresponding IIA string amplitudes. For nonperturbative computation beyond the planar level in the matrix model, techniques discussed in [30,[55][56][57][58] would be useful.…”

Section: Discussionmentioning

confidence: 99%

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

Nishigaki

Sugino

2014

J. High Energ. Phys.

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We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painlevé II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons.

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“…Apart from these universal regimes, when one finetunes the potential so that ρ(λ) ∼ λ 2m , a discrete series of multicritical points appear (with scaling 1/N 1/(2m+1) ). The behaviour on the scale of eigenvalue spacing in this regime is very difficult to extract [8,10,11]. A different (but also discrete) class of multicritical models appears when adding an appropriate fixed matrix to the random matrix.…”

Section: Universality Regimesmentioning

confidence: 99%

“…[8,10,11]. One can formulate a differential equation satisfied by the wavefunction ψ n (x), but taking the scaling limit is extremely difficult and e.g.…”

Section: Behaviour Near the Origin -Orthogonal Polynomialsmentioning

confidence: 99%

“…One can formulate a differential equation satisfied by the wavefunction ψ n (x), but taking the scaling limit is extremely difficult and e.g. for the case of quartic multicritical ensemble involves data from an auxiliary Painlevé equation [8,10]. Moreover the starting point of such considerations requires a polynomial potential, which is not the case for our class of models.…”

Section: Behaviour Near the Origin -Orthogonal Polynomialsmentioning

confidence: 99%

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New multicritical random matrix ensembles

Janik¹

2002

Nuclear Physics B

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In this paper we construct a class of random matrix ensembles labelled by a real parameter α ∈ (0, 1), whose eigenvalue density near zero behaves like |x| α . The eigenvalue spacing near zero scales like 1/N 1/(1+α) and thus these ensembles are representatives of a continuous series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the former case we obtain formulas for the eigenvalue density, while in the latter case we obtain approximate expressions for the scaling functions in the microscopic limit using a very simple approximate method based on the location of zeroes of orthogonal polynomials.

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Multicritical microscopic spectral correlators of hermitian and complex matrices

Cited by 61 publications

References 49 publications

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

New multicritical random matrix ensembles

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