The Tracy-Widom distribution at large Dyson index
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arXiv
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2025
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| author | Comtet, Alain Doussal, Pierre Le Smith, Naftali R. |
| author_facet | Comtet, Alain Doussal, Pierre Le Smith, Naftali R. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Tracy-Widom distribution at large Dyson index Comtet, Alain Doussal, Pierre Le Smith, Naftali R. Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics 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. |
| title | The Tracy-Widom distribution at large Dyson index |
| topic | Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics |
| url | https://arxiv.org/abs/2510.14433 |