Complete analysis of the high-fugacity phase

Analyze the high-fugacity phase of the AGT instanton gas for instanton number ν≫c≫1 beyond the leading string-like approximation, including the full competition among Young-tableau configurations and the resulting behavior near the true singularity z=1.

Background

The paper analyzes the low-fugacity phase in the c→∞ limit and examines the z→1 endpoint of the high-fugacity phase using string-like Young tableaux. It argues that string-like configurations remain relevant when ν≫c, while two-dimensional tableaux appear subdominant based primarily on numerical observations and scaling estimates.

A complete treatment of the high-fugacity phase—including the full configuration space, finite-c effects, and the precise dominant saddle structure—is not carried out. Such an analysis is needed to establish the phase’s dynamics rather than infer them from asymptotic representative classes.

References

It is beyond the scope of this work to perform a complete analysis -- we leave this to the future.

— Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence  (2608.13246 - Xu et al., 13 Aug 2026) in Section 5, “High fugacity phase”

In the future, it is interesting to study whether a limiting theory emerges at infinite order in 1/c, in which the mechanism for non-perturbative effects become more transparent.

— Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence  (2608.13246 - Xu et al., 13 Aug 2026) in Section 6, “Essential singularity at c=∞”