TheGUEAndBesselFunctions - crowlogic/arb4j GitHub Wiki

Gaussian Unitary Ensemble and Bessel Functions

The Gaussian Unitary Ensemble (GUE) is a class of random matrix ensembles that play a crucial role in various areas of mathematics and physics. The GUE consists of Hermitian matrices, i.e., square matrices that are equal to their own conjugate transpose. The probability distribution for a particular matrix is proportional to the exponential of the negative of the trace of the square of that matrix.

The relationship between the GUE and Bessel functions emerges when studying the eigenvalue statistics of random matrices from the GUE. Notably, the joint probability distribution for the eigenvalues possesses a specific structure that allows the problem to be transformed into an analysis of certain interacting particles. For large matrices, the distribution of the spacings between these "particles" (which correspond to the eigenvalues) is given by the "sine kernel," closely related to Bessel functions.

Particularly, the distribution of the largest eigenvalue of a GUE matrix relates to the Tracy-Widom distribution, which involves the Airy function, a special case of the Bessel function. If we denote the largest eigenvalue as $L$ and apply a suitable scaling and centering to the distribution, the limiting distribution as the size of the matrix approaches infinity is given by:

$$P(L < s) = e^{- \int_{s}^{\infty} (x-s)q^{2}(x) dx}$$

where $q(x)$ solves the Painlevé II differential equation:

$$q''(x) = xq(x) + 2q^{3}(x).$$

The function $q(x)$ can be expressed in terms of Airy functions, a specific form of Bessel functions. Therefore, while the GUE doesn't involve Bessel functions in its definition directly, they appear when examining the properties of the eigenvalues of GUE matrices.

The explicit solution to the Painlevé II equation mentioned above is:

$$q(x) = \frac{1}{2} \sqrt{-x} \frac{\mathrm{Ai}'(-x^{2/3})}{\mathrm{Ai}(-x^{2/3})}$$

where $\mathrm{Ai}(x)$ is the Airy function. The complexity of this topic necessitates a deep comprehension of random matrix theory, differential equations, and special functions.