Orador: Yang Chen, Departamento de Matemática, Universidade de Macau

Título: Smallest Eigenvalue of Large Hankel Matrices at Critical Point: comparing a Conjecture with parallelized computation

Data, hora e local: 14 de abril de 2021, às 16h, na plataforma Zoom no endereço

Resumo/abstract: We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned Hankel matrices. It is based on the LDLT decomposition and involves finding a k×k sub-matrix of the inverse of the original N×N Hankel matrix H^{-1}_N. 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 constitutes a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight $w(x)=e−^{x^{\beta}}$, supported on $[0,\infty[$ and $\beta>0$. Such weight generates a Hankel determinant, a fundamental object in random matrix theory. In the situation where $\beta>1/2$, the smallest eigenvalue tends to $0$ exponentially fast. If $\beta<1/2$, which is the situation where the classical moment problem is indeterminate, then the smallest eigenvalue is bounded from below by a positive number. If $\beta=1/2$, it is conjectured that the smallest eigenvalue tends to $0$ algebraically, with a precise exponent. The algorithm run on the HPCC producing a fantastic match between the theoretical value of $2/\pi$ and the numerical result.


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