关于行列式单位减Hankel矩阵
On determinants identity minus Hankel matrix
摘要：
本文研究了当$N$大时行列式$\det(I_N - βH_N)$的渐近性，其中$H_N$是符号中有100个跳跃间断的Hankel矩阵$H$的$N\times N$限制，满足$\|H\|\leq 1$ .此外，我们假设$β\in\mathbb C$与$|β|<1$和$I_N$表示单位矩阵。我们确定的一阶渐近的$N\to\infty$等行列式，并表明，他们表现出类似幂的渐近行为，指数取决于跳跃的高度。例如，对于具有矩阵元素$π^{-1}(j+k+1)^{-1}$的希尔伯特矩阵$\mathbf{H}$的$N \times N$截断，其中$j,k\in \mathbb Z_+$，我们获得$$ \log \det(I_N - β\mathbf{H}_N) = -\frac{\log N}{2π^2} \big(π\arcsin(β)+\arcsin^2(β)+o(1)\big),\qquad N\to\infty. $$
英文摘要：
In this note, we study the asymptotics of the determinant $\det(I_N - βH_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq 1$. Moreover, we assume $β\in\mathbb C$ with $|β|<1$ and $I_N$ denotes the identity matrix. We determine the first order asymtoptics as $N\to\infty$ of such determinants and show that they exhibit power-like asymptotic behaviour, with exponent depending on the height of the jumps. For example, for the $N \times N$ truncation of the Hilbert matrix $\mathbf{H}$ with matrix elements $π^{-1}(j+k+1)^{-1}$, where $j,k\in \mathbb Z_+$ we obtain $$ \log \det(I_N - β\mathbf{H}_N) = -\frac{\log N}{2π^2} \big(π\arcsin(β)+\arcsin^2(β)+o(1)\big),\qquad N\to\infty. $$
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