Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions

Understand

For theories that exhibit second order phase transition, we conjecture that the large-order asymptotic behavior of the strong-coupling ( High-Temperature) series expansion takes the form $\sigma^{n} n^{b}$ where $b$ is a universal parameter.

• The associated critical exponent is then given by $b+1$.
• The series itself can be approximated by the hypergeometric approximants $_{p}F_{p-1}$ which can mimic the same large-order behavior of the given series.
• Near the tip of the branch cut, the hypergeometric function $_{p}F_{p-1}$ has a power-law behavior from which the critical exponent and critical coupling can be extracted.