Asymptotic Expansion of β Matrix Models in the One-cut Regime

Borot, Gaëtan; Guionnet, Alice

doi:10.1007/s00220-012-1619-4

Cited by 177 publications

(444 citation statements)

References 49 publications

(72 reference statements)

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“…This fluid-phase condition should be compared with the so-called one-cut assumption for the 2D-OCP on the line [10]. Note that, strictly speaking, the 1/N -expansion also breaks down in Hermitian matrix models with a hard constraint of type λ max ≤ x (known as 'hard wall'), λ max being the largest eigenvalue.…”

Section: Discussionmentioning

confidence: 99%

“…This correction is provided by the second term in (10), where the short-distance cut-off (z) can be chosen as (z) −1 = √ N (dμ(z)/dz). A second contribution to the action A arises from the Jacobian J of the change of integration variables from the microscopic variables z k 's to μ:…”

Section: Thermodynamics Of the 2d-ocpmentioning

confidence: 99%

“…Note that the entropic term at order 1/N in the action vanishes at inverse temperature β = 4. The action representation (9) have recently proved that, under regularity assumptions on the equilibrium measure μ eq , the coefficient of the next-to-leading term is universal and equal to C β = −β/4 (the universality of C β was already known for the 2D-OCP constrained on the line [10,16,28]). In Sect.…”

Section: Thermodynamics Of the 2d-ocpmentioning

confidence: 99%

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Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

2016

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The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant βq 2 (where q is the particles charge and β the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number N of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N . We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine 1/N -expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for βq 2 = 2.

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Section: Discussionmentioning

confidence: 99%

Section: Thermodynamics Of the 2d-ocpmentioning

confidence: 99%

Section: Thermodynamics Of the 2d-ocpmentioning

confidence: 99%

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Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

2016

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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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“…Furthermore we will generally require the winding number of w(ζ) about ζ = 0 to vanish 4) however even this can be relaxed within our formalism, after the inclusion of additional boundary terms. Our ensemble is defined simply through the eigenvalue probability density function…”

Section: Jhep02(2015)173mentioning

confidence: 99%

Loop equation analysis of the circular β ensembles

Witte

Forrester

2015

J. High Energ. Phys.

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We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N . Using matching arguments for the resolvent functions of linear statistics f (ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment m k to an equivalent length. The leading large N , large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

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Asymptotic Expansion of β Matrix Models in the One-cut Regime

Cited by 177 publications

References 49 publications

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

Formal multidimensional integrals, stuffed maps, and topological recursion

Loop equation analysis of the circular β ensembles

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