Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls

Cunden, Fabio Deelan; Facchi, Paolo; Ligabò, Marilena; Vivo, Pierpaolo

doi:10.1007/s10955-019-02281-9

Cited by 13 publications

(11 citation statements)

References 61 publications

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“…This holds both in the symmetric and the more general setting: they belong to the same universality class. See [37,38] for detailed discussion on the universality of the third order phase transition in presence of a hard wall. We also stress that the discrete topology further constrains the eigenvalue density, but the condition is satisfied for all values of the parameters [9,10].…”

Section: 41mentioning

confidence: 99%

Exact equivalences and phase discrepancies between random matrix ensembles

Santilli,

Tierz

2020

Preprint

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We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions. We show that the models, while having the same exact evaluation for fixed values of the parameter, may present a different phase structure. We find phase transitions of the second and third order, depending on the model. Other relationships, via direct mapping, between the unitary matrix models and continuous random matrix ensembles on the real line, of Cauchy-Romanovski type, are presented and studied both exactly and asymptotically. The case of orthogonal and symplectic groups is studied as well and related to Wronskians of Chebyshev polynomials, that we evaluate at large N . Contents 1. Introduction 1 2. Random matrix ensembles on the unit circle 5 3. Random matrix ensembles on the real line 14 4. Outlook 23 Appendix A. Solution to saddle point equations on the unit circle 24 Appendix B. Free energies 31 Appendix C. Solution to the saddle point equations on the real line 33 References 37

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Section: 41mentioning

confidence: 99%

Exact equivalences and phase discrepancies between random matrix ensembles

Santilli,

Tierz

2020

Preprint

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“…This holds both in the symmetric and the more general setting: they belong to the same universality class. See [40,41] for detailed discussion on the universality of the third order phase transition in presence of a hard wall. We also stress that the discrete topology further constrains the eigenvalue density, but the condition is satisfied for all values of the parameters [9,10].…”

Section: 41mentioning

confidence: 99%

Exact equivalences and phase discrepancies between random matrix ensembles

Santilli¹,

Tierz²

2020

J. Stat. Mech.

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We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions. We show that the models, while having the same exact evaluation for fixed values of the parameter, may present a different phase structure. We find phase transitions of the second and third order, depending on the model. Other relationships, via direct mapping, between the unitary matrix models and continuous random matrix ensembles on the real line, of Cauchy-Romanovski type, are presented and studied both exactly and asymptotically. The case of orthogonal and symplectic groups is studied as well and related to Wronskians of Chebyshev polynomials, that we evaluate at large N . Contents 1. Introduction 1 2. Random matrix ensembles on the unit circle 5 3. Random matrix ensembles on the real line 14 4. Outlook 24 Appendix A. Solution to saddle point equations on the unit circle 25 Appendix B. Free energies 32 Appendix C. Solution to the saddle point equations on the real line 34 References 38

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“…Although we have derived the kernel (4.24) indirectly via the weak non-Hermiticity limit at the origin, we conjecture it to be universal, after an appropriate shift of the weight away from the origin plus rescalings. Because the appropriate Mehler or Poisson formula for the kernel (3.11) is lacking, when extending the sum to infinity 7 , we have been unable to directly take the strong non-Hermiticity limit.…”

Section: Weak Non-hermiticity In the Bulkmentioning

confidence: 99%

“…The case of a hard wall imposed at the edge of the droplet has been studied in [6]. When forcing the gas away from its equilibrium position, phase transitions may occur, see [7] for more general potentials and the general situation in d dimensions. Likewise, when β = 2c/N → 0 at fixed c, a smooth transition to a Gaussian is observed [8], including the weakly attractive case c ∈ (−2, 0].…”

Section: Introductionmentioning

confidence: 99%

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

Nagao

Akemann

Kieburg

et al. 2020

J. Phys. A: Math. Theor.

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We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature β = 2 we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all k-point correlation functions for a fixed number of particles N . These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-N limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine-and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

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Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls

Cited by 13 publications

References 61 publications

Exact equivalences and phase discrepancies between random matrix ensembles

Exact equivalences and phase discrepancies between random matrix ensembles

Exact equivalences and phase discrepancies between random matrix ensembles

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

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