Pade approximants of random Stieltjes series

TL;DR

This paper investigates the convergence properties of Pade approximants for a class of random Stieltjes functions defined by random continued fractions with gamma-distributed parameters, providing explicit formulas for convergence rates and pole distributions.

Contribution

It introduces a novel analysis of the convergence of Pade approximants for random Stieltjes functions using the Dyson-Schmidt method and explicit formulas involving Bessel functions.

Findings

Explicit formulas for almost-sure convergence rates.

Distribution of poles of the approximants derived.

Connection between random continued fractions and disordered systems established.

Abstract

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Pade approximants of the function S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional disordered system, and the results of Marklof et al. (2005), we obtain explicit formulae (in terms of modified Bessel functions) for the almost-sure rate of convergence of these approximants, and for the almost-sure distribution of their poles.