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Hard to soft edge transition for the Muttalib-Borodin ensembles with integer parameter $θ$

Dong Wang, Shuai-Xia Xu

TL;DR

This work analyzes the Muttalib-Borodin ensemble with integer parameter $\theta\ge 2$ in the transition regime at the left edge, establishing universal limiting kernels that interpolate between the hard-edge Meijer G kernels and the soft-edge Airy kernel. Using a vector-valued Riemann-Hilbert framework, the authors construct a size $(\theta+1)$ model RH problem at the origin, prove its solvability via a vanishing lemma, and extract limit functions $\phi^{(\tau)}$ and $\tilde{\phi}^{(\tau)}$ that encode the universal kernel $K^{(\tau)}(x,y)$. The kernel limit is given by $K^{(\tau)}(x,y) = \frac{\theta}{2\pi} x^{\alpha} \int_{\tau}^{\infty} \phi^{(\sigma)}(x) \tilde{\phi}^{(\sigma)}(y) \, d\sigma$ and, as $\tau\to-\infty$ or $+\infty$, reduces to the Meijer G or Airy kernels respectively, establishing a continuous transition universality. For $\theta=2$, a Lax pair analysis connects the local RH problem to the Chazy-I equation, which admits a Painlevé IV reduction and links to Drinfeld-Sokolov hierarchies. The results hold for a broad class of potentials $V$, demonstrating a robust integrable structure behind edge transitions in biorthogonal ensembles. This provides a unified, universal description of edge statistics in MB ensembles across hard-to-soft transitions and enriches the integrable systems perspective on determinantal processes.

Abstract

We find the universal limiting correlation kernels of the Muttalib-Borodin (MB) ensembles with integer parameter $θ\geq 2$ at $0$ in the transitive regime between the hard edge regime and the soft edge regime. This generalizes the previously studied hard edge to soft edge transition in unitarily invariant random matrix theory by Its, Kuijlaars and Östensson, which is the $θ= 1$ special case of our MB ensemble. The derivation is based on the vector Riemann-Hilbert (RH) problems for the biorthogonal polynomials associated with the MB ensemble. In the analysis of the RH problems, we construct matrix-valued model RH problems of size $(θ+ 1) \times (θ+ 1)$, and prove the solvability of the model RH problems by a vanishing lemma. The new limiting correlation kernels are proved to be universal for a large class of potential functions, and they interpolate the Meijer G-kernels for the hard edge regime and the Airy kernel for the soft edge regime. We observe that the new limiting correlation kernels have the integrability that is not seen in previous studies in random matrix theory and determinantal point processes. In the $θ= 2$ case, we give a detailed analysis of the Lax pair associated with the model RH problem, show that it results in the Chazy I equation, which has a Painlevé IV reduction, and find that the Lax pair is in the Drinfeld-Sokolov hierarchies.

Hard to soft edge transition for the Muttalib-Borodin ensembles with integer parameter $θ$

TL;DR

This work analyzes the Muttalib-Borodin ensemble with integer parameter in the transition regime at the left edge, establishing universal limiting kernels that interpolate between the hard-edge Meijer G kernels and the soft-edge Airy kernel. Using a vector-valued Riemann-Hilbert framework, the authors construct a size model RH problem at the origin, prove its solvability via a vanishing lemma, and extract limit functions and that encode the universal kernel . The kernel limit is given by and, as or , reduces to the Meijer G or Airy kernels respectively, establishing a continuous transition universality. For , a Lax pair analysis connects the local RH problem to the Chazy-I equation, which admits a Painlevé IV reduction and links to Drinfeld-Sokolov hierarchies. The results hold for a broad class of potentials , demonstrating a robust integrable structure behind edge transitions in biorthogonal ensembles. This provides a unified, universal description of edge statistics in MB ensembles across hard-to-soft transitions and enriches the integrable systems perspective on determinantal processes.

Abstract

We find the universal limiting correlation kernels of the Muttalib-Borodin (MB) ensembles with integer parameter at in the transitive regime between the hard edge regime and the soft edge regime. This generalizes the previously studied hard edge to soft edge transition in unitarily invariant random matrix theory by Its, Kuijlaars and Östensson, which is the special case of our MB ensemble. The derivation is based on the vector Riemann-Hilbert (RH) problems for the biorthogonal polynomials associated with the MB ensemble. In the analysis of the RH problems, we construct matrix-valued model RH problems of size , and prove the solvability of the model RH problems by a vanishing lemma. The new limiting correlation kernels are proved to be universal for a large class of potential functions, and they interpolate the Meijer G-kernels for the hard edge regime and the Airy kernel for the soft edge regime. We observe that the new limiting correlation kernels have the integrability that is not seen in previous studies in random matrix theory and determinantal point processes. In the case, we give a detailed analysis of the Lax pair associated with the model RH problem, show that it results in the Chazy I equation, which has a Painlevé IV reduction, and find that the Lax pair is in the Drinfeld-Sokolov hierarchies.

Paper Structure

This paper contains 60 sections, 25 theorems, 327 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Model Riemann-Hilbert problem and the limit functions $\phi^{(\tau)}$ and $\tilde{\phi}^{(\tau)}$
  4. Strategy and organization
  5. Notations
  6. Equilibrium measure in transition regime and soft edge regime
  7. Equilibrium measure in transition regime without deformation
  8. Deformation of the equilibrium measure: algebraic setup
  9. Deformation along $C_1$ when $\lvert t - 1 \rvert$ is small
  10. Deformation along $C_2$ when $1 - t$ is small positive
  11. Deformation along $C_2$ when $t \in (0, 1)$
  12. Asymptotic analysis of the RH problems for $p_n$ and $q_n$ in the transition regime
  13. Asymptotic analysis of the RH problem for $p_n$
  14. Transformations $Y \to T \to S \to Q$, and local parametrix around $b$
  15. Local parametrix around $0$
  16. ...and 45 more sections

Key Result

Theorem 1.5

Suppose $V$ satisfies Regularity conditions reg:primary and enu:reg:primary:2. In addition, suppose $A_2 \neq 0$, where $A_2$ is determined by $V$ through eq:defn_A2. Let $\tau$ be in a compact subset of $\mathbb{R}$, $x$ be in a compact subset of $\mathbb{C}$, and $y$ be in a compact subset of $\ov Let that depends on $n$. Let $p_j(x) = p^{(V_t)}_{n, k}(x)$, $q_k(x) = q^{(V_t)}_{n, k}(x)$ and $\

Figures (7)

  • Figure 1: The equilibrium measure of the MB ensemble with $\theta = 2$ and $V(x) = x^2 + \rho x$ with $\rho = 0$, $-1.8$, $-2$ and $-2.5$. When $\rho = 0$ and $-1.8$, the ensemble is in the hard edge regime in the sense that the density of the equilibrium measure blows up like $x^{-1/3}$. When $\rho = 0$, it is proved in Wang-Zhang21 that the left-most particles are around $0$ and their limiting distribution is given by the Meijer G-kernel. It is expected to hold also when $\rho = -1.8$. When $\rho = -2.5$, the density of the equilibrium measure vanishes like a square root at the left-end point of the support of the equilibrium measure, and we expect the left-most particles to be there and their limiting distribution to be the Tracy-Widom distribution given by the Airy kernel. When $\rho = -2$, the density of the equilibrium measure vanishes at the new speed of $x^{1/3}$ at $0$. Courtesy of Tom Claeys.
  • Figure 2: The jump contour for the RH problem \ref{['rhp:U']} for $U$ and for the RH problem \ref{['rhp:tildeU']} for $\widetilde{U}$.
  • Figure 3: The jump contour for RH problem \ref{['RHP:general_model']} for $\Phi$, for RH problem \ref{['RHP:general_model_tilde']} for $\widetilde{\Phi}$, and for the RH problem \ref{['RHP:general_model_tilde_check']} for $\check{\Phi}$.
  • Figure 4: Jump contour $\hat{\Sigma}^R$.
  • Figure 5: Jump contour $\hat{\pmb{\mathit{\Sigma}}}$. The jump is trivial for $\hat{\mathcal{R}}(s)$ on the dashed part.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.9
  • Lemma 1.10
  • Theorem 1.11
  • Remark 1.12
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.5
  • ...and 25 more