# Allan Trinh (2018-2021)

>## Finite N corrections in random matrix theory

### Abstract:

Many limit laws arise from the spectral theory of large random matrices. Complementary to this is the establishment of their rate of convergence. For example, the global scaled spectral density for a general class of large and cnetred \( N \times N \) real Wigner matrices is given by the Wigner semi-circular law with a rate of convergence \( O(1/N) \). A contribution of this thesis is to perform explicit calculations for the next leading finite \( N \) corrections, which are adapted from known results on this class of random matrices.

Limit laws also apply to the soft and hard edge scaling regimes at the spectrum edge. For classical unitary ensembles, such as the Gaussian and Laguerre ensembles, spectral statistics and their corresponding finite \( N \) corrections in soft and hard edge scaling variables follow from the asymptotic results derived from orthogonal polynomial theory. Earlier studies have shown the optimal rate of convergence is \( O(1/N^{2/3}) \) at the soft edge and \( O(1/N^2) \) at the hard edge. This problem is taken up to the \( \beta \) generalisation of the Gaussian and Laguerre ensembles. Using multidimensional integral formulas and the generalised hypergeometric functions based on Jack polynomial theory, calculations from the spectral density for even \( \beta \) show that, with a modified soft/hard edge scaling, the optimal rate of convergence -- in keeping with the unitary case -- is \( O(1/N^2) \) at the hard edge. In the case of the Laguerre \( \beta \) ensemble for general \( \beta > 0 \), similar methods are applied to evaluate finite \( N \) corrections to the distribution of the smallest eigenvalue for \( a \in \mathbb Z_{\ge 0} \), which again show an optimal rate of convergence \( O(1/N^2) \).

In our final study, we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble. For the classical Dyson indices \( \beta = 1,2,4 \), with use of the Routh-Romanovski polynomials, we show that this correction term is related to the leading term by a derivative operation. Using methods of analysis based on multi-dimensional hypergeometric functions, this structural property of the large \( N \) asymptotic expansion is shown to hold for the spectral density with \( \beta \) even. This implies that the optimal rate of convergence is \( O(1/N^2 )\), as found at the hard edge.

#### Published papers:

Comment on "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" (with P.J. Forrester), Phys. Rev. E **99**, 036101 (2019)

Optimal soft edge scaling variables for the Gaussian and Laguerre \( \beta \) ensembles (with P.J. Forrester), Nucl. Phys. B **938**, 621-639 (2019)

Finite size corrections at the hard edge for the Laguerre \( \beta \) ensemble (with P.J. Forrester), Stud. Appl. Math. **143**, 315-336 (2019)

Asymptotic correlations with corrections for the circular Jacobi \( \beta \) ensemble (with P.J. Forrester and S.-H. Li), J. Approx. Theory. **271**, 105633 (2021)