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(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:

$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$

Let $x_n = (x_{n,1}, \dots, x_{n, n})^{T}$ be the column vector satisfying that $A_n x_n = (1, \dots, 1)^{T}$. Here $T$ means taking the transpose of row vectors.

I tried numerical computation and conjecture that

$$ \lim_{n \to \infty} \frac{\log x_{n,1}}{\log n} = -\frac{1}{4}.$$

However I do not see how to prove this.

(ii) Consider a Toeplitz matrix $A_n = (a^{(n)}_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:

$$ a^{(n)}_{i, j} = \log \left(n/|i-j|\right), \ \text{ if } i \ne j; \ \ \ a^{(n)}_{i, j} := 1 + \log n, \text{ if } i = j. $$

I tried numerical computation and conjecture that

$$ \lim_{n \to \infty} \frac{\log x_{n,1}}{\log n} = -\frac{1}{2}.$$

I do not see how to prove this either.

A motivation for considering these questions comes from a problem in probability theory.

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  • $\begingroup$ I cannot help you, but I offer two further conjectures (which you may already have made yourself, of course): first, $\log(x_{n,1})\sim -\frac{1}{4}\log n-1$. second, the numbers $x_{n,i}^{-4}$ have a parabolic dependence on $i$. $\endgroup$
    – Marcel
    Commented Oct 26, 2017 at 12:24
  • $\begingroup$ Maybe this article might help P. Deift, A. Its, I. Krasovsky, "Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities", Ann. of Math. 174 (2011), 1243-1299. $\endgroup$
    – Nemo
    Commented Oct 26, 2017 at 16:23

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