Critical Hermitian matrix model with external source and Boussinesq hierarchy

TL;DR

This work analyzes the local eigenvalue statistics of a $2n$-dimensional Hermitian matrix model with external source and quartic potential, focusing on the origin under critical double-scaling limits. It introduces a novel $3 imes3$ model Riemann–Hilbert problem linked to the Boussinesq hierarchy, and constructs new limiting kernels $K^{( ext{γ,α,ρ,σ,τ})}$ that govern the cusp and multi-critical behavior; the α-parameter deforms the classical Pearcey kernel, recovering it when α=0. At the Pearcey point, the ext-source kernel converges to a deformed Pearcey kernel, while at the multi-critical point the limiting kernel is built from the second member of the Boussinesq hierarchy. The analysis connects the random matrix model to generalized Muttalib–Borodin ensembles, provides universality statements for Wigner-type matrices with external source, and links to the two-matrix Ising/2D gravity framework, yielding integrable structures via Lax pairs and Painlevé/Chazy reductions. These results expand the universality class of critical kernels in random matrix theory and suggest broader applicability of the Boussinesq-based RH framework to singular limits in external-source models.

Abstract

We consider the random Hermitian matrix model of dimension $2n$, with external source, defined by the probability density function \begin{equation*} \frac{1}{Z_{2n}} \lvert \det(M) \rvert^α e^{-2n\mathrm{Tr} (V(M) - AM)}, \quad V(x) = \frac{x^4}{4} - t\frac{x^2}{2}, \end{equation*} where the external source $A$ has two eigenvalues $\pm a$ of equal multiplicity. We investigate the limiting local statistics of the eigenvalues of $M$ around $0$ in certain critical regimes as $n \to \infty$. When the parameters $t$ and $a$ lie on a critical curve along which the limiting mean eigenvalue density vanishes as $|x|^{1/3}$, the double scaling limit of the correlation kernel is constructed from functions associated with the Boussinesq equation. This new limiting kernel reduces to the classical Pearcey kernel when $α= 0$. Furthermore, in the multi-critical case where the limiting mean eigenvalue density vanishes as $|x|^{5/3}$, the limiting kernel is built from the second member of the Boussinesq hierarchy. We derive the results by transforming the random matrix model into biorthogonal ensembles that are analogous to the Muttalib-Borodin ensemble, and then analyzing its asymptotic behavior via a vector Riemann-Hilbert problem.

Figures (3)

• Figure 1: The phase diagram of $t$ and $a$. Adapted from Bleher-Delvaux-Kuijlaars10.
• Figure 2: Shape of $\Sigma$ and the lens.
• Figure 3: The contour $\pmb{\mathit{\Sigma}}$ for the RH problem for $\mathcal{R}^{(n + k, j)}$. The curves $I_1(\Sigma^R_1)$, $I_1(\Sigma^R_2)$, $I_2(\Sigma^R_1)$, $I_2(\Sigma^R_2)$, $I_1(-\Sigma^{R, '}_1)$ and $I_1(-\Sigma^{R, '}_2)$ are labelled.

Theorems & Definitions (33)

• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Proposition 1
• Proposition 2