Research ArticleNo Access

A matrix model of a non-Hermitian $β$ $β$ -ensemble

Francesco Mezzadri

https://orcid.org/0000-0002-4572-5617

School of Mathematics, University of Bristol, Fry Building, Woodland Road Bristol BS8 1UG, UK

E-mail Address: f.mezzadri@bristol.ac.uk

Corresponding author.

Search for more papers by this author

and

Henry Taylor

https://orcid.org/0000-0002-3012-8342

School of Mathematics, University of Bristol, Fry Building, Woodland Road Bristol BS8 1UG, UK

E-mail Address: ht17630@bristol.ac.uk

Search for more papers by this author

https://doi.org/10.1142/S2010326324500278Cited by:0 (Source: Crossref)

Abstract

We introduce the first random matrix model of a complex $β$ $β$ -ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $β$ $β$ -ensembles discovered by [I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847]. The main feature of the model is that the exponent $β$ $β$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $ℝ_{+} .$ However, when $β = 2$ , the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.

Keywords:

AMSC: 15, 60

References

1. G. Akemann, A. Mielke and P. Päßler , Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-hermitian random matrices at noninteger $β$ , Phys. Rev. E 106 (2022) 014146. Crossref, Google Scholar
2. A. Altland and M. R. Zirnbauer , Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142–1161. Crossref, Google Scholar
3. S. Armstrong and S. Serfaty , Local laws and rigidity for Coulomb gases at any temperature, Ann. Prob. 49 (2021) 46–121. Crossref, Google Scholar
4. P. Bourgade, L. Erdős and H. T. Yau , Bulk universality of general $β$ -ensembles with non-convex potential, J. Math. Phys. 53 (2012) 095221. Crossref, Google Scholar
5. P. Bourgade, L. Erdős and H. T. Yau , Universality of general $β$ -ensembles, Duke Math. J. 163 (2014) 1127–1190. Crossref, Google Scholar
6. J. Breuer, P. J. Forrester and U. Smilansky , Random discrete Schrödinger operators from random matrix theory, J. Phys. A: Math. Theor. 40 (2007) F161–F168. Crossref, Google Scholar
7. S.-S. Byun and P. J. Forrester , Progress on the Study of the Ginibre Ensembles, KIAS Springer Series in Mathematics, Vol. 3 (Springer Nature, Singapore, 2025). Crossref, Google Scholar
8. E. Dueñez, Random matrix ensembles associated to compact symmetric spaces, Ph.D. thesis, Princeton University (2001). Google Scholar
9. E. Dueñez , Random matrix ensembles associated to compact symmetric spaces, Commun. Math. Phys. 244 (2004) 29–61. Crossref, Google Scholar
10. I. Dumitriu and A. Edelman , Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847. Crossref, Google Scholar
11. I. Dumitriu and A. Edelman , Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847. Crossref, Google Scholar
12. F. J. Dyson , Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140–156. Crossref, Google Scholar
13. F. J. Dyson , Statistical theory of the energy levels of complex systems. II, J. Math. Phys. 3 (1962) 157–165. Crossref, Google Scholar
14. F. J. Dyson , Statistical theory of the energy levels of complex systems. III, J. Math. Phys. 3 (1962) 166–175. Crossref, Google Scholar
15. F. J. Dyson , The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3 (1962) 1199–1215. Crossref, Google Scholar
16. A. Edelman and B. D. Sutton , From random matrices to stochastic operators, J. Stat. Phys. 127 (2007) 1121–1165. Crossref, Google Scholar
17. P. J. Forrester , Log-gases and Random Matrices (Princeton University Press, 2010). Crossref, Google Scholar
18. P. J. Forrester , Dyson’s disordered linear chain from a random matrix theory viewpoint, J. Math. Phys. 62 (2021) 103302. Crossref, Google Scholar
19. P. J. Forrester and E. M. Rains , Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Relat. Fields 131 (2005) 1–61. Crossref, Google Scholar
20. P. J. Forrester and E. M. Rains , Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices, Int. Math. Res. Not. 2006 (2006) 4836. Google Scholar
21. Y. V. Fyodorov and H.-J. Sommers , Random matrices close to Hermitian or unitary: Overview of methods and results, J. Phys. A: Math. Gen. 36 (2003) 3303–3347. Crossref, Google Scholar
22. J. Ginibre , Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6 (1965) 440–449. Crossref, Google Scholar
23. T. Grava, M. Gisonni, G. Gubbiotti and G. Mazzuca , Discrete integrable systems and random Lax matrices, J. Stat. Phys. 190 (2023) 1–35. Crossref, Google Scholar
24. T. Grava and G. Mazzuca , Generalized Gibbs ensemble of the Ablowitz–Ladik lattice, Circular $β$ -Ensemble and double confluent Heun equation, Commun. Math. Phys. 399 (2023) 1689–1729. Crossref, Google Scholar
25. R. Killip and I. Nenciu , Matrix models for circular ensembles, Int. Math. Res. Not. 2004 (2004) 2665–2701. Crossref, Google Scholar
26. M. Krishnapur, B. Rider and B. Virág , Universality of the stochastic Airy operator, Commun. Pure Appl. Math. 69 (2016) 145–199. Crossref, Google Scholar
27. S. N. Majumdar, A. Pal and G. Schehr , Extreme value statistics of correlated random variables: A pedagogical review, Phys. Rep. 840 (2020) 1–32. Crossref, Google Scholar
28. M. L. Mehta , Random Matrices, 3rd edn. (Elsevier Inc., 2004). Google Scholar
29. S. R. Nodari and S. Serfaty , Renormalized energy equidistribution and local charge balance in 2D Coulomb systems, Int. Math. Res. Not. 2015 (2015) 3035–3093. Google Scholar
30. B. N. Parlett , The Symmetric Eigenvalue Problem (SIAM, 1998). Crossref, Google Scholar
31. E. Sandier and S. Serfaty , From the Ginzburg–Landau model to vortex lattice problems, Commun. Math. Phys. 313 (2012) 635–743. Crossref, Google Scholar
32. E. Sandier and S. Serfaty , 2D Coulomb gases and the renormalized energy, Ann. Probab. 43 (2015) 2026–2083. Crossref, Google Scholar
33. H. Spohn , Generalized Gibbs ensembles of the classical Toda chain, J. Stat. Phys. 180 (2020) 4–22. Crossref, Google Scholar
34. H. Spohn, Hydrodynamic equations for the Toda lattice, (2021), arXiv:2101.06528v1. Google Scholar
35. H. F. Trotter , Eigenvalue distributions of large Hermitian matrices; Wigner’s semi-circle law and a theorem of Kac, Murdock, and Szegő, Adv. Math. 54 (1984) 67–82. Crossref, Google Scholar
36. M. R. Zirnbauer , Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys. 37 (1996) 4986–5018. Crossref, Google Scholar