Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities
The asymptotics when \(n\rightarrow \infty\) for \(n\)-dimensional Toeplitz, Hankel and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities are studied. A symbol with Fisher-Hartwig singularities is a function \(f(z)\) on the unit circle such that \(f(z) = e^{V(z)} z^{\sum_{j=0}^m \beta_j} \prod_{j=0}^m |z-z_j|^{2\alpha_j}g_{z_j,\beta_j}(z) z_j^{-\beta_j}\), \(z=e^{i\theta}\), \(\theta\in[0,2\pi]\), \(z_j=e^{i\theta_j}\), \(j=0,\dots,m\), \(0=\theta_0< \theta_1<\dots <\theta_m<2\pi\), \(g_{z_j,\beta_j}(z)=e^{i\pi\beta_j}\) if \(0\leq \mathrm{arg} z\leq \theta _j\), and \(e^{-i\pi\beta_j}\) if \(\theta_j\leq arg z\leq 2\pi\), \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\beta_j\in\mathbb{C}\), \(V(z)\) is a sufficiently smooth function on the unit circle. Let \(f_j=\frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta}d\theta\) be the Fourier coefficients of \(f\). Then the Toeplitz determinant with the symbol \(f\) is defined as \(D_n(f)=\det(f_{j-k})_{j,k=0}^{n-1}\), the Hankel determinant with the symbol \(f\) is defined as \(D_n(f)=\det(f_{j+k+1})_{j,k=0}^{n-1}\), and determinant of types \(D_n(f)=\det(f_{j-k}+f_{j+k})_{j,k=0}^{n-1}\), \(D_n(f)=\det(f_{j-k}-f_{j+k+2})_{j,k=0}^{n-1}\), \(D_n(f)=\det(f_{j-k}\pm f_{j+k+1})_{j,k=0}^{n-1}\) are called the Toeplitz+Hankel determinant. The determinants of such types corresponding to symbols with Fisher-Hartwig singularities have applications in statistical mechanics [\textit{M. E. Fisher} and \textit{R. E. Hartwig}, ``Toeplitz determinants: some applications, theorems, and conjectures'', Adv. Chem. Phys. 15, 333--353 (1969)], in random matrix theory [\textit{M. L. Mehta}, Random matrices. Rev. and enlarged 2. ed. Boston, MA: Academic Press, Inc. (1991; Zbl 0780.60014)], and even in number theory [\textit{É. Brézin} (ed.) et al., Application of random matrices in physics. NATO Science Series II: Mathematics, Physics and Chemistry 221. Dordrecht: Springer. (2006; Zbl 1120.82001)]. The first formula for the asymptotics of Toeplitz determinants in the case \(f\in L_1(S^1)\) was obtained by \textit{G. Szegö}, [``On certain Hermitian forms associated with the Fourier series of a positive function. Meddel. Lunds Univ. Mat. Sem. Suppl.-band M. Riesz, 228--238 (1952; Zbl 0048.04203)]. The situation when \(f\) has a Fisher-Hartwig singularity was considered by many authors: Widom (the case \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\beta_j=0\)) [\textit{H. Widom}, ``Toeplitz determinants with singular generating functions'', Am. J. Math. 95, 333--383 (1973; Zbl 0275.45006)], \textit{E. Basor} (the case \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\mathrm{Re} \beta_j=0\)) [``Asymptotic formulas for Toeplitz determinants'', Trans. Am. Math. Soc. 239, 33--65 (1978; Zbl 0409.47018)], \textit{A. Böttcher} and \textit{B. Silbermann} (the case \(|\mathrm{Re}\alpha_j|<\frac{1}{2}\), \(|\mathrm{Re} \beta_j|<\frac{1}{2}\)) [``Toeplitz matrices and determinants with Fisher-Hartwig symbols'', J. Funct. Anal. 63, 178--214 (1985; Zbl 0592.47016)], \textit{T. Ehrhardt} (the case \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\max_{jk}|\mathrm{Re}\beta_j-\beta_k|<1\)) [``A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities'', Basel: Birkhäuser. Oper. Theory, Adv. Appl. 124, 217--241 (2001; Zbl 0993.47028)]. Ehrhardt's theorem is presented in the paper (Theorem 1.1). The first main result of the paper under review is the formula for the asymptotics of the Hankel determinants in the case \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\beta_j\in\mathbb{C}\) (Theorem 1.13). The second main result (Theorem 1.20) is the asymptotics for the Hankel determinants of Hankel determinants, whose symbols \(w(x)\) satisfy the following condition. \(w(x)=e^{U(x)}\prod_{j=0}^{r+1}|x-\lambda_j|^{2\alpha_j}\omega_j(x)\), \(x\in[-1,1]\), \(1=\lambda_0>\lambda_1>\dots>\lambda_{r+1}=1\), \(\beta_0=\beta_{r+1}=0\), \(\mathrm{Re}\alpha_j>-\frac{1}{2}\), \(\mathrm{Re}\beta_j\in(-\frac{1}{2},\frac{1}{2})\), \(\omega_j(x)=e^{i\pi\beta_j)}\) if \(\mathrm{Re}x\leq\lambda_j\), and \(\omega_j(x)=e^{-i\pi\beta_j)}\) if \(\mathrm{Re}x>\lambda_j,\) \(U(x)\) is a sufficiently smooth function. Finally the asymptotics for the Toeplitz+Hankel determinants are obtained (Theorem 1.25). This theorem is proved for a symbol \(f(z)\) which possesses Fisher-Hartwig singularities, such that \(\theta_{r+1}=\pi\), \(\mathrm{Re}\beta_j\in (-\frac{1}{2},\frac{1}{2})\), \(j=1,\dots,r\), \(\beta_0=\beta_{r+1}=0\). An important role in the paper is played by an analysis of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
scientific article