Ferromagnet–spin-glass boundary and zero-field susceptibility

Prove that the ferromagnet–spin-glass boundary of the Sherrington–Kirkpatrick model with ferromagnetic interaction satisfies gamma_c([[? or derive the equivalent zero-field susceptibility identities for the Sherrington–Kirkpatrick model, including Toulouse’s marginality identity and the global inequality characterizing gamma_c([[? .

Background

The limiting free energy of the SKFI model is expressed as a variational problem over the magnetization, and the onset of ferromagnetic order is defined by γc(β)=inf⁡{γ>0:0∉Ω(β,γ)}\gamma_c(\beta)=\inf\{\gamma>0:0\notin\Omega(\beta,\gamma)\}. The paper proves several upper bounds on this boundary but does not determine it in the spin-glass temperature range β>1\beta>1.

The predicted boundary γc(β)=β\gamma_c(\beta)=\beta is related to the zero-field susceptibility φβ(h)=FSK(β,h)−FSK(β,0)\varphi_\beta(h)=F^{SK}(\beta,h)-F^{SK}(\beta,0). Establishing the boundary requires, in particular, resolving the mathematical status of Toulouse’s marginality identity and the associated global bound on φβ(h)\varphi_\beta(h).

References

The variational structure~eq:variational identifies \gamma_c with a zero-field susceptibility of the SK model: writing \varphi_\beta(h) := F{SK}(\beta,h)-F{SK}(\beta,0), a convexity argument shows \gamma_c(\beta) = \inf_{h>0}\frac{h2}{2\varphi_\beta(h)}, so \gamma_c(\beta)=\beta is equivalent to the pair of statements \lim_{h\downarrow0}2\varphi_\beta(h)/h2 = 1/\beta (Toulouse's marginality identity, an established consequence of the Parisi solution in the physics literature but open mathematically) and \varphi_\beta(h) h2/(2\beta) for all h>0.

— Fluctuations of the free energy of the Sherrington-Kirkpatrick model with ferromagnetic interaction  (2608.25362 - Dey et al., 26 Aug 2026) in Remark 2.1, “The ferromagnet–spin glass boundary” (Section 1.2, immediately following Proposition 1.1)