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Flowing to normality and the fate of the single ring theorem

Joshua Feinberg1, Roman Riser2,3, Richard Scalettar4, and A. Zee5

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Phys. Rev. E 114, 044106 – Published 5 October, 2026

DOI: https://doi.org/10.1103/mjdb-pjxm

Abstract

Random non-Hermitian matrix ensembles with double-sided rotation invariance obey, in the limit of large matrix size, the single ring theorem, which states that the support of the mean eigenvalue distribution in the complex plane is either a disk or an annulus. In contrast, rotational-invariant random normal matrix ensembles can have mean eigenvalue densities supported over any number of concentric annuli in the complex plane. In this paper we introduce and investigate, both analytically and numerically, a non-Hermitian matrix model which flows from a generic matrix distribution obeying the single ring theorem to a distribution of normal matrices by tuning a parameter which penalizes non-normality. We observe numerically breakdown of the single ring theorem as the model flows towards normality and determine the critical value of the parameter at which the transition occurs. We also study in detail the behavior of the singular values of these matrices under the flow. These singular values form a Fermi gas confined to the positive half-line. In particular, we find that at small values of the flow parameter, the interparticle spacings in the gas exhibit Wigner-Dyson repulsion, whereas for asymptotically large values of the flow parameter, at the normal matrix end point of the flow, the spacing statistics is Poissonian. The flow interpolates continuously between these two types of statistics. However, this change in statistics is not related directly to breaking of the single ring theorem, which occurs very early-on along the flow, in the regime of Wigner-Dyson statistics. Finally, we introduce a certain ensemble of random permutations associated with the gas, and make a conjecture on how to use it in order to reconstruct approximately the average density of complex eigenvalues from that of the singular values in the large-N limit.

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References (62)

  1. J. Feinberg and A. Zee, Non-Gaussian non-Hermitian random matrix theory: Phase transition and addition formalism, Nucl. Phys. B 501, 643 (1997).
  2. J. Feinberg, R. Scalettar, and A. Zee, ‘Single ring theorem’ and the disk-annulus phase transition, J. Math. Phys. 42, 5718 (2001).
  3. J. Feinberg, Non-Hermitian random matrix theory: Summation of planar diagrams, the ‘single ring’ theorem and the disc-annulus phase transition, J. Phys. A: Math. Theor. 39, 10029 (2006).
  4. Y. Wei and Y. V. Fyodorov, On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values, J. Phys. A: Math. Theor. 41, 502001 (2008).
  5. Y. V. Fyodorov and B. A. Khoruzhenko, Averages of spectal determinants and ‘single ring theorem’ of Feinberg and Zee, Acta Phys. Polon. B: Math. Theor. 38, 4067 (2007).
  6. A. Guionnet, M. Krishnapur, and O. Zeitouni, The single ring theorem, Ann. Math. 174, 1189 (2011).
  7. S. Belinschi, M. A. Nowak, R. Speicher, and W. Tarnowski, Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem, J. Phys. A: Math. Theor. 50, 105204 (2017).
  8. M. A. Nowak and W. Tarnowski, Complete diagrammatics of the single ring theorem, Phys. Rev. E 96, 042149 (2017).
  9. Y. Ameur, C. Charlier, and J. Cronvall, The two-dimensional Coulomb gas: Fluctuations through a spectral gap, Arch. Ration. Mech. Anal. 249, 63 (2025).
  10. Sampling over infinite realizations of ϕ would result in perfect rotational invariant shape. Slight possible deviations from perfect rotational invariance in Fig. 2 are due to finite sampling.
  11. M. Moshe, H. Neuberger, and B. Shapiro, Generalized ensembles of random matrices, Phys. Rev. Lett. 73, 1497 (1994).
  12. J. Feinberg and A. Zee, Renormalizing rectangles and other topics in random matrix theory, J. Stat. Phys. 87, 473 (1997).
  13. A. Anderson, R. C. Myers, and V. Periwal, Branched polymers from a double-scaling limit of matrix models, Nucl. Phys. B 360, 463 (1991).
  14. C. Itzykson and J.-B. Zuber, The planar approximation. II, J. Math. Phys. 21, 411 (1980).
  15. W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (John Wiley & Sons, New York, 1968), Vol. 1.
  16. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (John Wiley & Sons, New York, 1971), Vol. 2.
  17. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier, Amsterdam, 2004).
  18. G. Livan, M. Novaes, and P. Vivo, Introduction to Random Matrices—Theory and Practice (Springer Nature, Berlin, 2018).
  19. R. Riser and E. Kanzieper, Power spectrum and form factor in random diagonal matrices and integrable billiards, Ann. Phys. 425, 168393 (2021).
  20. G. Parisi, String theory on the one dimensional lattice, Phys. Lett. B 238, 213 (1990).
  21. E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Planar diagrams, Commun. Math. Phys. 59, 35 (1978).
  22. S. Chadha, G. Mahoux, and M. L. Mehta, A method of integration over matrix variables: II, J. Phys. A: Math. Gen. 14, 579 (1981).
  23. E. Brézin, V. A. Kazakov, and Al. B. Zamolodchikov, Scaling violation in a field theory of closed strings in one physical dimension, Nucl. Phys. B 338, 673 (1990).
  24. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, Inc., New York, 1965).
  25. L. S. Schulman, Techniques and Applications of Path Integration (Dover, Mineola, NY, 2005).
  26. M. Ndawana and V. Kravtsov, Energy level statistics of a critical random matrix ensemble, J. Phys. A: Math. Gen. 36, 3639 (2003).
  27. V. Kravtsov, O. Yevtushenko, and E. Cuevas, Level compressibility in a critical random matrix ensemble: The second virial coefficients, J. Phys. A: Math. Gen. 39, 2021 (2006); corrigendum: J. Phys. A: Math. Theor. 44, 189501 (2011).
  28. J. Chalker, V. Kravtsov, and I. Lerner, Spectral rigidity and eigenfunction correlations at the Anderson transition, Jetp Lett. 64, 386 (1996).
  29. R. J. Szabo, Equivariant Cohomology and Localization of Path Integrals (Springer Verlag, Berlin, 2000).
  30. R. J. Szabo, Equivariant localization of path integrals, arXiv:hep-th/9608068.
  31. A. M. Vershik and S. Kerov, Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl. 15, 246 (1981).
  32. A. M. Vershik and S. Kerov, Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group, Func. Anal. Appl. 19, 21 (1985).
  33. B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. Math. 26, 206 (1977).
  34. F. F. Abraham, Computational statistical mechanics methodology, applications and supercomputing, Adv. Phys. 35, 1 (1986).
  35. G. Bhanot, The Metropolis algorithm, Rep. Prog. Phys. 51, 429 (1988).
  36. S. Chib and E. Greenberg, Understanding the Metropolis-Hastings algorithm, Am. Stat. 49, 327 (1995).
  37. W. Krauth, Statistical Mechanics: Algorithms and Computations (Oxford University Press, Oxford, 2006), Vol. 13.
  38. Theoretically, for finite N, the density is positive everywhere since (18) is always nonzero for any configuration as long as all SSV are mutually distinct. But in the appearing gap, the density is exponentially suppressed, such that even for rather small N the probability to find a SSV there becomes vanishingly small.
  39. M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).
  40. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 5th ed. (Cambridge University Press, Cambridge, UK, 2021).
  41. R. H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 (1987).
  42. H. G. Evertz, G. Lana, and M. Marcu, Cluster algorithm for vertex models, Phys. Rev. Lett. 70, 875 (1993).
  43. R. T. Scalettar, R. M. Noack, and R. R. P. Singh, Ergodicity at large couplings with the determinant Monte Carlo algorithm, Phys. Rev. B 44, 10502 (1991).
  44. B. Cohen-Stead, O. Bradley, C. Miles, G. Batrouni, R. Scalettar, and K. Barros, Fast and scalable quantum Monte Carlo simulations of electron-phonon models, Phys. Rev. E 105, 065302 (2022).
  45. B. Rider, A limit theorem at the edge of a non-Hermitian random matrix ensemble, J. Phys. A: Math. Gen. 36, 3401 (2003).
  46. M. Bender, Edge scaling limits for a family of non-Hermitian random matrix ensembles, Probab. Theory Relat. Fields 147, 241 (2010).
  47. G. Cipolloni, L. Erdős, and D. Schröder, Edge universality for non-Hermitian random matrices, Probab. Theory Relat. Fields 179, 1 (2021).
  48. A. Campbell, G. Cipolloni, L. Erdős, and H. C. Ji, On the spectral edge of non-Hermitian random matrices, Ann. Probab. 53, 2256 (2025).
  49. G. Akemann, Y. V. Fyodorov, and D. V. Savin, Spectral density and eigenvector nonorthogonality in complex symmetric random matrices, arXiv:2511.21643.
  50. D.-Z. Liu and L. Zhang, Repeated erfc statistics for deformed GinUEs, in Annales Henri Poincaré (Springer, Berlin, 2026), Vol. 27, pp. 1165–1205.
  51. Y. Ameur, H. Hedenmalm, and N. Makarov, Random normal matrices and Ward identities, Ann. Probab. 43, 1157 (2015).
  52. S.-Y. Lee and R. Riser, Fine asymptotic behavior for eigenvalues of random normal matrices: Ellipse case, J. Math. Phys. 57, 023302 (2016).
  53. Y. Ameur, N.-G. Kang, and N. Makarov, Rescaling Ward identities in the random normal matrix model, Construct. Approx. 50, 63 (2019).
  54. H. Hedenmalm and A. Wennman, Planar orthogonal polynomials and boundary universality in the random normal matrix model, Acta Math. 227, 309 (2021).
  55. J. Cronvall and A. Wennman, A direct approach to soft and hard edge universality for random normal matrices, arXiv:2511.18628.
  56. C. Charlier, Smallest gaps of the two-dimensional Coulomb gas, arXiv:2507.23502.
  57. M. S. S. Challa, D. P. Landau, and K. Binder, Finite-size effects at temperature-driven first-order transitions, Phys. Rev. B 34, 1841 (1986).
  58. J. Lee and J. M. Kosterlitz, Finite-size scaling and Monte Carlo simulations of first-order phase transitions, Phys. Rev. B 43, 3265 (1991).
  59. L.-L. Chau and Y. Yu, Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects, Phys. Lett. A 167, 452 (1992).
  60. More exotic self-adjoint extensions leading to conserved nonvanishing persistent probability current are irrelevant for our discussion and are therefore excluded.
  61. E. Farhi and S. Gutmann, The functional integral on the half-line, Int. J. Mod. Phys. A 05, 3029 (1990).
  62. T. E. Clark, R. Menikoff, and D. H. Sharp, Quantum mechanics on the half-line using path integrals, Phys. Rev. D 22, 3012 (1980).

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Phys. Rev. E 114, 044106– Published 5 October, 2026
  • Received 16 June 2026
  • Accepted 17 September 2026
Article part of CHORUS
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