Pade approximants of random Stieltjes series

Details Pade approximants of random Stieltjes series Pade approximants of random Stieltjes series

2007-08-09

arxiv.org/abs/0708.1245v1

Physics

Mathematical Physics

To appear in Proc Roy Soc

Scientific paper

10.1098/rspa.2007.0014

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.

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