Artigo Acesso aberto Revisado por pares

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

2007; Springer Science+Business Media; Volume: 27; Issue: 2 Linguagem: Inglês

10.1007/s00365-007-0675-z

ISSN

1432-0940

Autores

K. T-R McLaughlin, A. H. Vartanian, Xiaowen Zhou,

Tópico(s)

Matrix Theory and Algorithms

Resumo

Let $\Lambda^{{\Bbb R}}$ denote the linear space over ${\Bbb R}$ spanned by $z^{k}, k \in {\Bbb Z}$ . Define the (real) inner product $\langle \cdot,\cdot \rangle_{{\cal L}} : \Lambda^{{\Bbb R}} \times \Lambda^{{\Bbb R}} \to {\Bbb R}, (f,g) \mapsto \int_{{\Bbb R}}f(s)g(s) \exp (-{\cal N} V(s)) \, {\rm d} s, {\cal N} \in {\Bbb N}$ , where V satisfies: (i) V is real analytic on ${\Bbb R} \backslash \{0\}$ ; (ii) $\lim_{\vert x \vert \to \infty}(V(x)/{\rm ln} (x^{2} + 1)) = +\infty$ ; and (iii) $\lim_{\vert x \vert \to 0}(V(x)/{\rm ln} (x^{-2} + 1)) = +\infty$ . Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2},\ldots,z^{-k},z^{k},\ldots \rbrace$ with respect to $\langle \cdot,\cdot \rangle_{{\cal L}}$ yields the even degree and odd degree orthonormal Laurent polynomials $\lbrace \phi_{m}(z) \rbrace_{m=0}^{\infty}: \phi_{2n}(z) = \xi^{(2n)}_{-n}z^{-n} + \cdots + \xi^{(2n)}_{n}z^{n}, \xi^{(2n)}_{n} > 0$ , and $\phi_{2n+1}(z) = \xi^{(2n+1)}_{-n-1}z^{-n-1} + \cdots + \xi^{(2n+1)}_{n}z^{n}, \xi^{(2n+1)}_{-n-1} > 0$ . Define the even degree and odd degree monic orthogonal Laurent polynomials: $\pi_{2n}(z) := (\xi^{(2n)}_{n})^{-1} \phi_{2n}(z)$ and $\pi_{2n+1}(z) := (\xi^{(2n+1)}_{-n-1})^{-1} \phi_{2n+1}(z)$ . Asymptotics in the double-scaling limit ${\cal N},n \to \infty$ such that ${\cal N}/n = 1 + o(1)$ of $\pi_{2n+1}(z)$ (in the entire complex plane), $\xi^{(2n+1)}_{-n-1}$ , and $\phi_{2n+1}(z)$ (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ${\Bbb R}$ , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].

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