Summability of the perturbative expansion for a zero-dimensional disordered spin model

TL;DR

This paper proves that the perturbative expansion for the free energy of a zero-dimensional disordered Ising model is Borel-summable within certain parameters, using a two-step summation process, supported by numerical calculations.

Contribution

It introduces a novel two-step summation method demonstrating Borel-summability of the perturbative series in a disordered spin model.

Findings

Perturbative series is Borel-summable in a specific parameter range.

Two-step summation improves convergence of the series.

Numerical calculations validate the analytical results.

Abstract

We show analytically that the perturbative expansion for the free energy of the zero dimensional (quenched) disordered Ising model is Borel-summable in a certain range of parameters, provided that the summation is carried out in two steps: first, in the strength of the original coupling of the Ising model and subsequently in the variance of the quenched disorder. This result is illustrated by some high-precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method.