The multiplicative constant in asymptotics of higher-order analogues of the Tracy-Widom distribution

TL;DR

This work constructs and analyzes higher-order analogues of the Tracy-Widom distribution arising from unitary random matrix ensembles near singular edge points, using kernels tied to the even Painlevé I hierarchy P_{I}^{2k}. By formulating and solving a Riemann-Hilbert problem for the associated kernels and employing Deift-Zhou steepest descent, the authors derive a complete large-gap asymptotic expansion for the logarithm of the Fredholm determinant, including a novel constant term expressed via an integral of the Hamiltonian h attached to a real pole-free P_{I}^{2k} solution. They prove the total integral of h vanishes and establish a transition from the higher-order TW distributions to the classical TW distribution in the appropriate asymptotic limit, while also providing explicit formulas for the constant C^{(k)} in terms of h and related functions. The methodology extends to other critical kernels in mathematical physics, illustrating a versatile approach to constant-term evaluation in integrable determinants with Painlevé structure.

Abstract

In this paper, we are concerned with higher-order analogues of the Tracy-Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by functions related to the even members of the Painlevé I hierarchy $\mathrm{P_{I}^{2k}}, k\in\mathbb{N}^{+}$, and are regarded as higher-order analogues of the Airy kernel. We present a novel approach to establish the multiplicative constant in the large gap asymptotics of the distribution, resolving an open problem in the work of Clayes, Its and Krasovsky. An important new feature of the expression is the involvement of an integral of the Hamiltonian associated with a special, real, pole-free solution for $\mathrm{P_{I}^{2k}}$. In addition, we show that the total integral of the Hamiltonian vanishes for all $k$, and establish a transition from the higher-order Tracy-Widom distribution to the classical one in the asymptotic regime. Our approach can also be adapted to calculate similar critical constants in other problems arising from mathematical physics.

Figures (16)

• Figure 1: The three regions in asymptotic studies of $\frac{\partial}{\partial s} F(s; x)$ and $\frac{\partial}{\partial x} F(s; x)$. The exponentially decay and algebraic growth regions are separated by the transitional region in the middle. The dashed curve stands for the critical curve $x = -\alpha_k s^{2k+1}$, where $\alpha_k$ is given in \ref{['eq-def-Ak']}.
• Figure 2: The contours of integration in the $(x,s)$-plane. We choose the red lines if $x_0>0$ and the blue lines if $x_0<0$. The dashed curve is the critical curve $x = -\alpha_k s^{2k+1}$, where $\alpha_k$ is given in \ref{['eq-def-Ak']}.
• Figure 4: Regions I--V and jump contours of the RH problem for $X$. The case $s<0$ is presented on the left and the case $s>0$ is presented on the right.
• Figure 5: The jump contour $\Sigma^{T}$ of the RH problem for $T$.
• Figure 6: The jump contour $\Sigma^{R}$ of the RH problem for $R$ in Section \ref{['sec:RH-analysis-case-I']}.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1: Large $s$ asymptotics of $\frac{\partial F}{\partial x}$ in the algebraic growth region
• Lemma 2.2: Large $x$ asymptotics of $\frac{\partial F}{\partial s}$ in the algebraic growth region
• Lemma 2.3: Large $x$ asymptotics of $\frac{\partial F}{\partial s}$ in the transition region
• Lemma 2.4: Large $x$ asymptotics of $\frac{\partial F}{\partial s}$ in the exponential decay region
• Remark 2.5
• Remark 3.1