Soft edge limit of the Laguerre beta-ensemble at the lower edge

Technical Summary: Soft Edge Limit of the Laguerre Beta-Ensemble at the Lower Edge

Problem Statement
The paper investigates the asymptotic behavior of the lower edge of the size $n$ Laguerre beta-ensemble with parameter $a = a_n$ as $n \to \infty$. The ensemble is defined by a joint probability density function involving a parameter $\beta > 0$ and a parameter $a > -1$. While the global spectral distribution is governed by the Marchenko-Pastur law, the microscopic behavior at the spectral edges requires specific scaling limits.

Previous work had established two distinct regimes for the lower edge:

1. Hard Edge: When $a$ is fixed ($a > -1$), the scaled eigenvalues converge to the stochastic Bessel process ($\text{Bessel}_{\beta, a}$).
2. Soft Edge (Upper and Lower with fixed ratio): When $a_n/n \to \gamma \in (0, \infty)$, the lower edge converges to the stochastic Airy process ($\text{Airy}_\beta$).

The gap in the literature concerned the regime where $a_n \to \infty$ but $a_n/n \to 0$. It was conjectured that in this intermediate regime, the lower edge should also converge to the $\text{Airy}_\beta$ process, completing the picture of possible edge scaling limits. This paper provides a rigorous proof of this conjecture.

Methodology
The authors employ the tridiagonal matrix representation of the Laguerre beta-ensemble introduced by Dumitriu and Edelman. The eigenvalues of the ensemble correspond to the eigenvalues of the random tridiagonal matrix $L_n L_n^\top$, where $L_n$ is a bidiagonal matrix with independent chi-distributed entries.

The proof strategy relies on analyzing the inverse of the rescaled and recentered matrix $M_n = L_n L_n^\top - \mu_n I_n$, where $\mu_n$ is the expected location of the lower edge. The authors view the inverse of this matrix as a discrete approximation of an integral operator. The goal is to show that this discrete operator converges in the Hilbert-Schmidt norm to the inverse of the stochastic Airy operator, $\text{Airy}_\beta^{-1}$.

The proof is divided into two regimes based on the growth rate of $a_n$:

1. Fast Growth Regime ($a_n \gg (\log \log n)^3$):

   • The authors analyze the partial solutions of the eigenvalue equation $M_n u = 0$.
   • They divide the analysis into three regimes based on the index $i$ of the vector components $u_i$:
     • Regime 1: Uses the classical theory of convergence of Markov chains to diffusions to show that the rescaled vector components converge to the solution of the stochastic Airy differential equation.
     • Regime 2: This is the most technically challenging part. The authors develop martingale fluctuation estimates to control the discrete Riccati transform associated with $u_i$. This requires the assumption that $a_n$ grows sufficiently fast to ensure the drift terms dominate the fluctuations.
     • Regime 3: Uses concentration bounds to prove that the vector components grow sufficiently fast, allowing the application of a discrete Wronskian identity to translate bounds on $u$ into bounds on the inverse matrix entries.
   • The convergence of the discrete kernel to the continuous kernel is established in the $L^2$ norm, implying norm-resolvent convergence.

2. Slow Growth Regime ($1 \ll a_n \le (\log n)^{1/2}$):

   • The direct operator analysis is difficult due to the slow growth of $a_n$.
   • Instead, the authors use a coupling argument. They establish a quantitative bound showing that the eigenvalues of the finite ensemble are close to the points of the stochastic Bessel process $\text{Bessel}_{\beta, 2a_n}$.
   • They then rely on a known result (Theorem D) regarding the transition from the hard edge to the soft edge: as the parameter of the Bessel process goes to infinity, the rescaled process converges to $\text{Airy}_\beta$.
   • A diagonal argument combines the finite ensemble convergence to the Bessel process with the Bessel-to-Airy transition to prove the result.

Key Contributions and Results

• Main Theorem (Theorem 1): The paper proves that for $a_n \to \infty$ and $a_n/n \to 0$, the scaled lower edge eigenvalues converge in distribution to the $\text{Airy}_\beta$ process. Specifically:
  $$a_n^{-4/3} n \left( \Lambda_n - (\sqrt{n + 2a_n} - \sqrt{n})^2 \right) \Rightarrow \text{Airy}_\beta$$
• Operator Level Convergence: For the fast growth regime ($a_n \gg (\log \log n)^3$), the authors establish operator-level convergence. They show that the inverse of the scaled tridiagonal matrix converges to the inverse of the stochastic Airy operator in the Hilbert-Schmidt norm. This is a stronger result than mere distributional convergence of eigenvalues.
• Generalization to Other Ensembles: The methods developed for the fast growth regime are applied to prove operator-level soft edge limits for:
  • The lower edge of the Gaussian beta-ensemble.
  • The lower edge of the Laguerre beta-ensemble when $\liminf a_n/n > 0$ (Theorem 3).
  • The upper edge of the Laguerre beta-ensemble (via symmetry and modification).

Significance
The paper claims to "complete the picture" of the possible edge scaling limits for the Laguerre beta-ensemble with fixed $\beta > 0$. By resolving the case where $a_n \to \infty$ but $a_n/n \to 0$, it bridges the gap between the hard edge (fixed $a$) and the soft edge (fixed ratio $a/n$) regimes.

The authors note that while the special case $\beta=2$ with $a_n = c\sqrt{n}$ was previously known via determinantal structures, this work provides the first proof for general diverging rates for $\beta=2$ and the first results for $\beta=1$ and $\beta=4$ in this regime. The operator-level convergence results extend the understanding of universality near the edge for a wide class of beta-ensembles, confirming that the stochastic Airy operator is the universal limit for soft edges under these conditions. The paper explicitly states that it does not provide a new proof for the hard-to-soft transition of the limiting processes themselves (Theorem D), but rather uses it as a crucial ingredient for the slow growth regime.