Square-root critical behavior conjecture for the formal two-matrix model

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Let F0F_0 denote the planar free energy of the formal two-matrix model referred to in the source, and let gcg_c be a critical point. The string susceptibility exponent is denoted by γ\gamma.

Square-root critical behavior conjecture. The formal matrix model has the asymptotic expansion

ddgF0∼2−21−g/gcas g→gc,\frac{d}{d g}F_{0} \sim 2-2 \sqrt{1-g/g_{c}} \quad\text{as }g\rightarrow g_{c},

which implies that γ=1/2\gamma=1/2.

The conjecture is motivated by bootstrap data, curve fitting, and the expected critical behavior, although the proposed algebraic ansatz for m2m_2 is ruled out by the series expansion. The status of the asserted critical behavior is not resolved in the supplied text.

References

Primary source

Masoud Khalkhali, Nathan Pagliaroli, Andrei Parfeni and Brayden Smith, “Bootstrapping the critical behavior of multi-matrix models”, arXiv:2409.07565 (2025).

Additional references

5 papers in this index state this conjecture (2008–2024). The statement above is taken from the most recent of them; the others are arXiv:2110.01118, arXiv:1408.1467, arXiv:1304.8011, arXiv:0801.3946.

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