The Tracy-Widom Distribution at Large Dyson Index

Abstract We study the Tracy-Widom (TW) distribution $$f_\beta (a)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in the limit of large Dyson index $$\beta \rightarrow +\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . This distribution describes the fluctuations of the rescaled largest eigenvalue $$a_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> of the Gaussian (alias Hermite) ensemble (G $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> E) of (infinitely) large random matrices. We show that, at large $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> , its probability density function takes the large deviation form $$f_\beta (a) \sim e^{-\beta \Phi (a)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . While the typical deviation of $$a_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> around its mean is Gaussian of variance $$O(1/\beta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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 $$\Phi (a)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\Phi (a)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> numerically for all a and compare with exact numerical computations of the TW distribution at finite $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . These results are obtained by applying saddle-point approximations to an associated problem of energy levels $$E=-a$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> , 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 $$a_1&gt;a_2&gt;\dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:m