Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2305.13184

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

Francesco Mezzadri·Henry Taylor
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the first random matrix model of a complex $β$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $β$-ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent $β$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $\mathbb{R}_+$. 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.

Explore related subjects

math-phmath.MPmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Francesco Mezzadri, Henry Taylor. 2024-11-08. A matrix model of a non-Hermitian $β$-ensemble. https://doi.org/10.1142/s2010326324500278

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model

We present a complete analysis of the glass transition in the self-overlap-corrected Sherrington\--Kirkpatrick (SK) model in a transverse magnetic field, also referred to as the quantum SK (QSK) model. In particular, we determine the phase boundary separating the glassy and paramagnetic phases explicitly. Such an analytic characterization of the glass transition is not expected for the true QSK and even unknown for most vector glass models. Despite being of independent interest, the analysis of the self-overlap corrected QSK model serves as important ingredient in the characterization of paramagnetic behavior in the real QSK. The proof is based on a simplified Parisi variational principle for the quantum pressure, which only involves classical Parisi order parameters. As part of the proof, we also analyze the pressure of the self-overlap-constrained quantum SK model and its Parisi description, as well as the pressure of generalized quantum Hopfield models.

math-ph↗
arXiv2026

How to Recover Oscillation-Free Pressure in Real Fluids: The RFQC Method and Its Liquid-Upwind Anomaly

From the perspective of continuum thermodynamics, we revisit the pressure oscillation problem in finite-volume methods for multiphase real fluids and clarify the physical counterpart of the Real Fluid Quasi-Conservative (RFQC) method. The pressure oscillation in conservative finite-volume methods originates from their implicit thermodynamic equilibrium assumption, whereas recovering an oscillation-free pressure requires additional physical information. The RFQC method achieves this by evolving the affine parameters xi and E0 of the isentropic internal-energy-pressure relation along pathlines, while the thermodynamic re-projection converts the deviation from the isentropic trajectory into an internal-energy error, thereby ensuring the thermodynamic consistency and numerical stability of the method. We then investigate the applicability limit of the RFQC method and identify a Liquid-upwind Anomaly (LUA) in extreme phase-change cases. For a Riemann problem involving liquid-vapor phase change, a numerical anomaly may occur if a liquid-upwind translational velocity is initially superimposed. Theoretical analysis reveals that this anomaly is initiated by the jump in the affine slope xi during phase change, which delays pressure rise in the downstream vapor cell. Concurrently, the re-projection removes the positive pressure increment, repeatedly generating large internal-energy errors and trapping the vapor cell in a cycle of delayed pressure recovery. The analysis indicates that the LUA is a start-up anomaly, which can be resolved by introducing a regularization strategy at the initial discontinuity. With the proposed regularization strategy, the RFQC method is equipped with enhanced accuracy and robustness for extreme thermodynamic flows, such as sonic phase-change jets.

math-ph↗
arXiv2026

Nonlocal Cubic Density Gibbs Measures from Bosonic Gibbs States with Three-Body Interactions

We study the high-temperature mean-field limit of grand-canonical bosonic Gibbs states on the torus with renormalized nonlocal three-body interactions. In dimensions two and three, we construct the limiting nonlinear classical Gibbs measure and prove convergence of the relative free energy and of the reduced density matrices of every fixed order; in three dimensions, a smallness condition on the interaction is imposed. The proof combines the density-channel representation of the interaction with a coherent-state variational method based on the upper-symbol representation of the free Gibbs state. The same framework also contains, as a special case, the homogeneous positive-type model studied by Lewin, Nam, and Rougerie (2021).

math-ph↗