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Asymptotic expansion of the hard-to-soft edge transition

Luming Yao, Lun Zhang

TL;DR

The paper addresses the universal hard-to-soft edge transition in Laguerre unitary ensembles by proving a full asymptotic expansion for the symmetrically transformed Bessel kernel $\hat{K}_{\nu}^{\mathrm{Bes}}$ in powers of $h_{\nu}=2^{-1/3}\nu^{-2/3}$. It employs a Riemann-Hilbert characterization of the Bessel kernel and a Deift-Zhou steepest descent analysis to derive a uniform expansion of the kernel in terms of the Airy kernel $K^{\mathrm{Ai}}$ plus explicit correction kernels $K_j(x,y)$ with polynomial coefficients in $x$ and $y$, i.e. $\hat{K}_{\nu}^{\mathrm{Bes}}(x,y)=K^{\mathrm{Ai}}(x,y)+\sum_{j=1}^{\mathfrak{m}} K_j(x,y)h_{\nu}^j+\mathcal{O}(h_{\nu}^{\mathfrak{m}+1})$. The method further yields a corresponding hard-edge expansion for the hard-edge gap probability $E_2^{\mathrm{hard}}$ and provides explicit expressions for the early correction terms, thereby confirming conjectures of Bornemann and facilitating higher-order edge corrections in random matrix theory and related combinatorial models. The results build a systematic framework to compute the polynomials $p_{j,\kappa\lambda}$ that define the correction kernels and demonstrate uniform validity across regimes where the transformed variables remain bounded or grow up to $h_{\nu}^{-1}$.

Abstract

By showing that the symmetrically transformed Bessel kernel admits a full asymptotic expansion for large parameter, we establish a hard-to-soft edge transition expansion. This resolves a conjecture recently proposed by Bornemann.

Asymptotic expansion of the hard-to-soft edge transition

TL;DR

The paper addresses the universal hard-to-soft edge transition in Laguerre unitary ensembles by proving a full asymptotic expansion for the symmetrically transformed Bessel kernel in powers of . It employs a Riemann-Hilbert characterization of the Bessel kernel and a Deift-Zhou steepest descent analysis to derive a uniform expansion of the kernel in terms of the Airy kernel plus explicit correction kernels with polynomial coefficients in and , i.e. . The method further yields a corresponding hard-edge expansion for the hard-edge gap probability and provides explicit expressions for the early correction terms, thereby confirming conjectures of Bornemann and facilitating higher-order edge corrections in random matrix theory and related combinatorial models. The results build a systematic framework to compute the polynomials that define the correction kernels and demonstrate uniform validity across regimes where the transformed variables remain bounded or grow up to .

Abstract

By showing that the symmetrically transformed Bessel kernel admits a full asymptotic expansion for large parameter, we establish a hard-to-soft edge transition expansion. This resolves a conjecture recently proposed by Bornemann.
Paper Structure (12 sections, 7 theorems, 117 equations, 5 figures)

This paper contains 12 sections, 7 theorems, 117 equations, 5 figures.

Table of Contents

  1. Introduction and statement of results
  2. An RH characterization of the Bessel kernel
  3. Asymptotic analysis of the RH problem for $\Psi$ with large $\nu$
  4. First transformation: $\Psi \to Y$
  5. Second transformation: $Y \to T$
  6. Third transformation: $T \to S$
  7. Global parametrix
  8. Local parametrix
  9. Final transformation
  10. Proof of Theorem \ref{['th:1']}
  11. The Airy parametrix
  12. Asymptotic expansion of the Bessel functions for large order and large argument

Key Result

Theorem 1.1

With $\hat{K}_{\nu}^{\mathrm{Bes}}(x,y)$ defined in def:inducedkernel, we have, for any $\mathfrak{m}\in\mathbb{N}$, uniformly valid for $t_0 \le x, y <h_{\nu}^{-1}$ with $t_0$ being any fixed real number. Preserving uniformity, the expansion can be repeatedly differentiated w.r.t. the variable $x$ and $y$. Here, $K^{\mathrm{Ai}}$ is the Airy kernel given in def:KAi and with $p_{j, \kappa \lambd

Figures (5)

  • Figure 1: The jump contours of the RH problem for $\Psi$.
  • Figure 2: The jump contour $\Gamma_T$ of the RH problem for $T$.
  • Figure 3: Image of $\mathrm{Re}\, g$: the solid line is the contour of $\mathrm{Re}\,{g(z)}=0$, the "$-$" sign is the region where $\mathrm{Re}\,{g(z)}<0$ and the "$+$" sign shows the region where $\mathrm{Re}\,{g(z)}>0$.
  • Figure 4: The jump contours of the RH problem for $R$.
  • Figure 5: The jump contours of the RH problem for $\Phi^{({\mathrm{Ai}})}$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 3.3
  • Proposition 3.7
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.3
  • ...and 4 more