On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

Albeverio, Sergio; Pastur, L. А.; Shcherbina, Mariya

doi:10.1007/s002200100531

Cited by 74 publications

(196 citation statements)

References 18 publications

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“…Then, the existence of the all-order asymptotic expansion at β = 2 was proven by Albeverio, Pastur and Shcherbina [1] by combining Schwinger-Dyson equations and the bounds derived in [47]. In particular, this work proved that the coefficients of the asymptotic expansion coincide with the formal generating series enumerating ribbon graphs of [34] also known under the name of "maps".…”

Section: The Schwinger-dyson Equations For a General Potentialmentioning

confidence: 80%

“…Then, in Appendix D, we derive an exact expression for the partition function Z N [V G ] when β = 1 and V G is a quadratic potential. We also obtain there the large-N asymptotics of Z N [V G ] up to o (1). This result is instrumental in deriving the asymptotic expansion of Z N [V] for more general potential, since the Gaussian partition function always appears as a factor of the latter.…”

Section: N · W(t N ξ)mentioning

confidence: 84%

“…The physical idea behind universality is that the logarithmic repulsion dictates the local behaviour of the particles 1 . The universality classes should only depend on β and the local environment of the chosen position on R. Typically one expects that, when N → +∞, the particles will localise on some union of segments ∪ k [a k ; b k ] and that, up to a O N −1 precision, the p th particle will localise around a "classical" position γ cl p .…”

Section: Local Fluctuations and Universalitymentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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Section: The Schwinger-dyson Equations For a General Potentialmentioning

confidence: 80%

Section: N · W(t N ξ)mentioning

confidence: 84%

Section: Local Fluctuations and Universalitymentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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“…In the convergent matrix model, the assumptions and technical steps are of different nature and are more involved, because one needs first to justify the existence of a large N expansion for an appropriate topology. In the more simple convergent model (1.1), the large N asymptotic expansion were established in the one-cut case in [APS01,BG12], and in the multi-cut case in [BG13] justifying the heuristics of [BDE00,Eyn09] under natural assumptions on V . The generalization of this approach to the model (1.2) seen as a convergent matrix model will be addressed in a subsequent work [BGK].…”

Section: Outlinementioning

confidence: 99%

Formal multidimensional integrals, stuffed maps, and topological recursion

Borot

2014

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [EO07a] with initial condition. In terms of a 1d gas of eigenvalues, this model includes -on top of the squared Vandermonde -multilinear interactions of any order between the eigenvalues. In this problem, the initial data pω In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps". In a sense, our results complete the program of the "moment method" initiated in the 90s to compute the formal 1{N in the one hermitian matrix model.

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“…We use (1.13) as a system of nonlinear equations for α (n) k and solve it by the perturbation theory method. To construct a zero order solution of the system (1.13), following the method of [1] and [13] we use the link of the OPUC with the unitary matrix models, which eigenvalue distribution has the form…”

Section: Introductionmentioning

confidence: 99%