Critical inverse-temperature threshold for the surface-sum representation

Determine whether the critical value $\upbeta^*$ at which the large-N surface-sum representation first ceases to hold in two dimensions equals $1/2$.

Background

The companion theorem establishing convergence to the surface-sum formula is proved only for sufficiently small inverse temperature, while the paper observes that the representation cannot remain valid for $\upbeta>1$. For a single plaquette, the proposed limiting spectral density becomes negative when $\upbeta>1/2$, suggesting a sharper threshold.

The authors conjecturally identify $1/2$ as the critical value in dimension two but do not prove it. They also leave open the possibility that large loops could exhibit a different critical value, potentially equal to one.

References

For this reason (the paper would seem to suggest the same conclusion), we suspect, at least in dimension two, that $\upbeta* = \frac{1}{2}$, and it would be nice to confirm whether this is true.

— Surface sums in two-dimensional large-$N$ lattice Yang--Mills: Cancellations and explicit computation for general loops  (2508.13827 - Borga et al., 19 Aug 2025) in Remark 1.7, “Remark: critical”