Extend the asymptotic results to arbitrary external magnetic fields

Establish the stated asymptotic behavior of the partition function for the Ising model on tetravalent planar maps with an external magnetic field for every magnetic-field parameter c>0, rather than only for c in a neighborhood of 1.

Background

The paper proves the asymptotic behavior of the partition-function coefficients only for weak external magnetic fields, represented by values of c sufficiently close to 1. This restriction is imposed for technical convenience in the singularity analysis of the generating functions and their discriminants.

The authors explicitly indicate that the same result is expected to hold for every c>0, but they do not provide a proof beyond the weak-field regime. Resolving this problem would establish the full external-field range without the neighborhood-of-1 restriction. as implied by the supplied text

References

Notice that our result is valid for values of c close to 1, meaning that we only consider weak external magnetic fields. This is only a technical restriction. The result should be true for all c>0, but we did not push our arguments for the sake of simplicity.

— Ising model with external magnetic field on random planar maps: Critical exponents  (2511.03688 - Tokka, 5 Nov 2025) in Section 1, immediately after Theorem 'Asymptotic behavior of the coefficients of Z'; revisited in Section 'Ising model with external magnetic field: cneq 1', opening paragraphs