Smallest Eigenvalue of Large Hankel Matrices at Critical Point: Comparing Conjecture With Parallelised Computation

Abstract

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the {\it LDLT} decomposition and involves finding a $k \times k$ sub-matrix of the inverse of the original $N \times N$ Hankel matrix $H_N^{-1}$ . The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitute a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight $w(x)={\rm e}^{-x^β},$ supported on $[0,\infty)$ and $β>0.$ Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where $β>1/2,$ the smallest eigenvalue tend to 0, exponentially fast as $N$ gets large. If $β<1/2,$ the situation where the classical moment problem is indeterminate, the smallest eigenvalue is bounded from below by a positive number for all $N$, including infinity. If $β=1/2,$ it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing fantastic match between the theoretical value of $2/π$ and the numerical result.