The Tracy-Widom distribution at large Dyson index

TL;DR

This work analyzes the Tracy-Widom distribution at large Dyson index β, showing that the distribution shrinks as fβ(a) ~ e^{-β Φ(a)} and that the rate function Φ(a) is obtained from a Painlevé II-type equation. It develops two complementary methods—the saddle-point approach applied to the stochastic Airy operator and a weak-noise (diffusion) analysis of the Riccati representation—to derive the full large-β behavior, including tails and typical fluctuations, and to compute cumulants up to order four. The results extend to the Airyβ point process, yielding large-deviation forms for marginal and joint edge eigenvalue distributions and gaps, with explicit asymptotics and numerical validation against finite-β TW distributions. The findings reveal a deep link between large-β edge statistics and integrable structures (Painlevé II) and offer a framework to study higher-order edge observables in β-ensembles and related stochastic operators.

Abstract

We study the Tracy-Widom (TW) distribution $f_β(a)$ in the limit of large Dyson index $β\to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble (G$β$E) of (infinitely) large random matrices. We show that, at large $β$, its probability density function takes the large deviation form $f_β(a) \sim e^{-βΦ(a)}$. While the typical deviation of $a_1$ around its mean is Gaussian of variance $O(1/β)$, this large deviation form describes the probability of rare events with deviation $O(1)$, and governs the behavior of the higher cumulants. We obtain the rate function $Φ(a)$ as a solution of a Painlevé II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute $Φ(a)$ numerically for all $a$ and compare with exact numerical computations of the TW distribution at finite $β$. These results are obtained by applying saddle-point approximations to an associated problem of energy levels $E=-a$, for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being $E$ (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process $a_1>a_2>\dots$ which describes all edge eigenvalues of the G$β$E, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of $a_i$, the joint distributions, and the gap distributions.

Figures (5)

• Figure 1: (a) Solid line: The large deviation function $\Phi(a_1)= s(E_1=-a_1)$ as a function of the largest eigenvalue $a_1 = -E_1$, where $E_1$ is the ground state energy of the potential \ref{['V0def']}. $s(E_1=-a_1)$ is computed on numerical solutions to Eq. \ref{['NonLinearPhiEq']}. Red dotted lines: the asymptotic behaviors (which include subleading corrections) \ref{['sSolRightTailSubleading']}, \ref{['SSolLeftTail']} and \ref{['sofECubicOrder']} (with $i=1$) of $s(a_1)$ at $a_1 \to \pm\infty$ and $a_1 \simeq \zeta_1$, respectively. Blue dotted lines: the leading-order asymptotic behaviors \ref{['sSolRightTail']} and \ref{['SSolLeftTailLeadingOrder']} at $a_i \to \pm\infty$, respectively. Also plotted are numerical computations of $-\frac{1}{\beta} \ln f_{\beta}(a_1)$ where $f_{\beta}(a_1)$ is the Tracy-Widom distribution, for relatively large Dyson indices, $\beta=10,20$ (dashed and dot-dashed lines, respectively). $f_\beta(a_1)$ is computed using the numerical method from Ref. BloemendalThesis. One can see that our theory describes the distributions very well. (b) similar to (a) but for the second largest eigenvalue $a_2 = -E_2$. In (b), the red dotted lines at $a_2 \to \pm\infty$ correspond to Eqs. \ref{['sSolRightTaili']} and \ref{['SSolLeftTailGenerali']} (with $i=2$).
• Figure 2: (a) Solid line: Exact (numerically-obtained) solution $\phi_1(x)$ to Eq. \ref{['NonLinearPhiEq']} for ground-state energy $E_1 = -126$. Dashed line: the (approximate) theoretical prediction \ref{['phiSolLocalized']}, where $x_0 = 0.415$ was initially obtained as a fitting parameter but then found to agree with our formula \ref{['x0ofEAsymptotic']}. (b) $x_0$ (which we determined numerically as the local maximum of $|\phi_1(x)|$) as a function of $E_1$ (markers), which show excellent agreement with the $E_1 \to -\infty$ analytic prediction \ref{['x0ofEAsymptotic']} (solid line).
• Figure 3: (a) Solid line: Exact (numerically-obtained) solution $\phi_1(x)$ to Eq. \ref{['NonLinearPhiEq']} for ground-state energy $E_1 = 12.6$. Dashed line: the (approximate) theoretical prediction \ref{['phiSolLargeE']}. (b) Similarly for the first excited state, with $E_2 = 17.6$.
• Figure 5: The time-dependent potential $U(z,t)$ as a function of $t$ for $t-E > 0$ (solid line) and $t-E < 0$ (dashed line), see Eq. \ref{['Uofzt']}. The fat dots correspond to the points $z=z_0$ and $z=z_1$ described in the text.
• Figure 6: (a) The Ablowitz-Segur solution $w(\tau)$ for a value of $k$ slightly less than $1$, i.e. $k\simeq1-5.88\times 10^{-6}$, corresponding to $E=6.3$, see also Ref. Rosales78. The solid line is the physical solution which we use here. The value of $-E$ can be read from the first negative zero. The dotted line is the (non-physical) continuation of the solution to $\tau < -E$. The dashed line corresponds to the Thomas-Fermi approximation \ref{['TF']}. (b) A translation of the Ablowitz Segur solution in terms of the function $v(t)$ (c) a sketch of the corresponding trajectory $q(t)$.