The problem originated in the open-questions list of Baik and Lee's analysis of the spherical Sherrington–Kirkpatrick (SSK) model. The SSK near-critical window β=1+O(log⁡N/N1/3)\beta=1+O(\sqrt{\log N}/N^{1/3}) and the Gaussian-to-Tracy–Widom crossover distribution that Baik and Lee left open have since been determined — the window scale and critical Gaussian limit by Landon (2022), and the full fixed-bb crossover law by Johnstone, Klochkov, Onatski and Pavlyshyn (2024), confirming Baik–Lee's conjecture. What survives, and constitutes this problem, is the item Baik and Lee stated verbatim: for the classical Ising SK model, 'a limit theorem for the fluctuations in the low temperature regime still remains as an open question. The limit theorem is not known even for the zero temperature case.' For β>βc=1\beta>\beta_c=1 the fluctuation exponent is bounded only by ρ≤−1/2\rho\le -1/2 and no limiting distribution is known; for the zero-temperature ground-state energy, competing conjectures (ρ=−3/4\rho=-3/4 and ρ=−5/6\rho=-5/6) are undecided, and no limiting distribution is known there either. A 2026 series of papers resolved only the critical point β=1\beta=1 itself. The problem asks for the order and the limiting law of the centered free energy throughout β>βc\beta>\beta_c, including the zero-temperature case.

Background

The Sherrington–Kirkpatrick (SK) model is the archetypal mean-field spin glass, introduced by Sherrington and Kirkpatrick in 1975 as a solvable model of a disordered magnet. It consists of NN Ising spins σi∈{±1}\sigma_i\in\{\pm 1\} with random pairwise interactions: the couplings gijg_{ij} (i<ji<j) are independent standard Gaussian random variables, and the Hamiltonian is HN(σ)=1N∑i<jgijσiσj.H_N(\sigma)=\frac{1}{\sqrt{N}}\sum_{i<j}g_{ij}\sigma_i\sigma_j. The free energy at inverse temperature β\beta is FN(β)=1Nlog⁡∑σ∈{±1}NeβHN(σ)F_N(\beta)=\frac{1}{N}\log\sum_{\sigma\in\{\pm1\}^N}e^{\beta H_N(\sigma)} (normalizations vary across the literature; with this one the critical inverse temperature is βc=1\beta_c=1), and the Gibbs measure is the random probability distribution on spin configurations with weights proportional to eβHN(σ)e^{\beta H_N(\sigma)}. The limit F(β)=lim⁡N→∞FN(β)F(\beta)=\lim_{N\to\infty}F_N(\beta) is given by the Parisi formula — a recursive variational expression, proved rigorously by Guerra, Talagrand and Panchenko — which pins down the equilibrium free energy but carries no information about its fluctuations.

Write the fluctuation exponent ρ\rho if the centered free energy FN(β)−EFN(β)F_N(\beta)-\mathbb{E}F_N(\beta) has typical size Nρ+o(1)N^{\rho+o(1)}. The high-temperature phase β<1\beta<1 is well understood: the centered free energy is asymptotically Gaussian on the N−1N^{-1} scale (variance Θ(N−2)\Theta(N^{-2})). The critical point itself was settled only in 2026: Du and Huang (arXiv:2607.02172) proved Var(FN(1))=16N−2log⁡N+O(N−2)\mathrm{Var}(F_N(1))=\frac{1}{6}N^{-2}\log N+O(N^{-2}) with a Gaussian central limit theorem — confirming the physics prediction recorded as Conjecture 2.4 in the 2025 survey of Collins-Woodfin and Le — and also identified the critical scale of the two-replica overlap R1,2=1N∑iσi(1)σi(2)R_{1,2}=\frac{1}{N}\sum_i\sigma_i^{(1)}\sigma_i^{(2)} (a measure of the correlation between two independent samples from the Gibbs measure); Chen (arXiv:2607.21392) independently proved the same variance asymptotics via a Gaussian-convexity route; Schertzer (arXiv:2606.21360) proved the weaker logarithmic bounds; and Dey and Kang (arXiv:2603.05636) treated the near-critical approach from the high-temperature side.

Nothing comparable is known for β>1\beta>1, the low-temperature (spin-glass) phase, where the Gibbs measure is expected to decompose into a hierarchically organized family of pure states (disjoint clusters of configurations dominating the Gibbs weights), the phenomenon known as replica symmetry breaking. The best rigorous statements are bounds: Chatterjee's superconcentration estimate — superconcentration denotes variance strictly below the generic C/NC/N bound that the Gaussian Poincaré inequality gives for functionals of independent Gaussians; here Var(FN(β))≤Cβ2log⁡(2+Cβ)/(Nlog⁡N)\mathrm{Var}(F_N(\beta))\le C\beta^2\log(2+C\beta)/(N\log N) valid at all finite β\beta, which in particular gives ρ≤−1/2\rho\le -1/2; his general lower-bound method, giving fluctuations at least of order N−1N^{-1} at all finite temperatures; and, at zero temperature — where the free energy degenerates to the ground-state energy GSN=max⁡σHN(σ)/N\mathrm{GS}_N=\max_{\sigma}H_N(\sigma)/N — the results that the fluctuations are o(N−1/2)o(N^{-1/2}) (Chen, Handschy and Lerman) and at least of order N−1N^{-1}. For the zero-temperature ground-state energy, physics predictions split between ρ=−3/4\rho=-3/4 and ρ=−5/6\rho=-5/6, two values that are numerically nearly indistinguishable at accessible system sizes, and no candidate limiting distribution has been proposed. The 2025 survey of Collins-Woodfin and Le records this as the central open problem of the area, stating that the fluctuations of the free energy in the low-temperature regime remain an open problem.

The question acquires its sharpest form by contrast with the spherical variant. In the spherical SK model of Kosterlitz, Thouless and Jones — the same random Hamiltonian but with spins constrained to the unit sphere — the partition function reduces to a function of the eigenvalues of a Gaussian orthogonal ensemble (GOE) random matrix (the ensemble of real symmetric matrices with independent Gaussian entries), so the free-energy fluctuations are known in every regime: Gaussian at high temperature, governed by the GOE Tracy–Widom law (the universal limiting distribution of the largest eigenvalue of large GOE matrices) at low temperature, and by an explicit Gaussian-to-Tracy–Widom crossover in the critical window β=1+bN−1/3log⁡N\beta=1+bN^{-1/3}\sqrt{\log N}, established by Landon and by Johnstone, Klochkov, Onatski and Pavlyshyn in work confirming the conjecture of Baik and Lee that opened this line of investigation. The Ising model admits no such random-matrix representation, and it was in that same paper — after solving the spherical problem — that Baik and Lee stated the question recorded here.

Problem Statement

For the Sherrington–Kirkpatrick model with i.i.d. standard Gaussian couplings and no external field, determine the fluctuations of the free energy throughout the low-temperature regime β>βc=1\beta>\beta_c=1, including the zero-temperature limiting case. Concretely:

  1. Identify the fluctuation exponent: prove that for each fixed β>1\beta>1 there is an exponent ρ(β)\rho(\beta) such that FN(β)−EFN(β)F_N(\beta)-\mathbb{E}F_N(\beta) has typical size Nρ(β)+o(1)N^{\rho(\beta)+o(1)} — for instance in the sense that Var(FN(β))=N2ρ(β)+o(1)\mathrm{Var}(F_N(\beta))=N^{2\rho(\beta)+o(1)}; and identify the exponent for the zero-temperature ground-state energy GSN=max⁡σHN(σ)/N\mathrm{GS}_N=\max_{\sigma}H_N(\sigma)/N, deciding between the conjectured values ρ=−3/4\rho=-3/4 and ρ=−5/6\rho=-5/6 or refuting both, and determine whether the fixed-β>1\beta>1 free-energy exponent takes the same value.

  2. Prove a limit theorem: after centering and scaling at the correct order, establish convergence in distribution of FN(β)F_N(\beta) for β>1\beta>1 and of GSN\mathrm{GS}_N, and identify the limiting law — in particular, whether it is Gaussian.

These two items are nested rather than independent: the exponent sought in item 1 is a prerequisite for the limit theorem of item 2, so establishing the exponent alone is a partial milestone, while a complete resolution requires the limit theorem of item 2. This is the question stated verbatim by Baik and Lee (2016): 'a limit theorem for the fluctuations in the low temperature regime still remains as an open question. The limit theorem is not known even for the zero temperature case.' Results restricted to the critical point β=1\beta=1 or to the high-temperature phase β<1\beta<1 do not address the problem.

References

  1. David Sherrington and Scott Kirkpatrick, "Solvable Model of a Spin-Glass", Physical Review Letters 35 (1975) 1792–1796, doi:10.1103/PhysRevLett.35.1792, https://doi.org/10.1103/PhysRevLett.35.1792
  2. J. M. Kosterlitz, D. J. Thouless, and Raymund C. Jones, "Spherical Model of a Spin-Glass", Physical Review Letters 36 (1976) 1217–1220, doi:10.1103/PhysRevLett.36.1217, https://doi.org/10.1103/PhysRevLett.36.1217
  3. Jinho Baik and Ji Oon Lee, "Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model", Journal of Statistical Physics 165 (2016) 185–224, doi:10.1007/s10955-016-1610-0, arXiv:1505.07349, https://arxiv.org/abs/1505.07349
  4. Benjamin Landon, "Free energy fluctuations of the two-spin spherical SK model at critical temperature", Journal of Mathematical Physics 63 (2022) 033301, doi:10.1063/5.0054298, arXiv:2010.06691, https://arxiv.org/abs/2010.06691
  5. Iain M. Johnstone, Yegor Klochkov, Alexei Onatski, and Damian Pavlyshyn, "Spin glass to paramagnetic transition and triple point in spherical SK model", Journal of Statistical Physics 191 (2024), Article 99, doi:10.1007/s10955-024-03296-7, arXiv:2104.07629, https://arxiv.org/abs/2104.07629
  6. Elizabeth Collins-Woodfin and Han Gia Le, "Free energy fluctuations in SK and related spin glass models: A literature survey", arXiv:2510.26960 (2025), https://arxiv.org/abs/2510.26960
  7. Sourav Chatterjee, "Disorder chaos and multiple valleys in spin glasses", arXiv:0907.3381 (2009), https://arxiv.org/abs/0907.3381
  8. Sourav Chatterjee, "A general method for lower bounds on fluctuations of random variables", Annals of Probability 47 (2019) 2140–2171, arXiv:1706.04290, https://arxiv.org/abs/1706.04290
  9. Wei-Kuo Chen, Madeline Handschy, and Gilad Lerman, "On the energy landscape of the mixed even p-spin model", Probability Theory and Related Fields 171 (2018) 53–95, arXiv:1609.04368, https://arxiv.org/abs/1609.04368
  10. Partha S. Dey and Taegu Kang, "Fluctuations for the Sherrington–Kirkpatrick spin glass model near the critical temperature", arXiv:2603.05636 (2026), https://arxiv.org/abs/2603.05636
  11. Adrien Schertzer, "The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited", arXiv:2606.21360 (2026), https://arxiv.org/abs/2606.21360
  12. Hang Du and Brice Huang, "Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature", arXiv:2607.02172 (2026), https://arxiv.org/abs/2607.02172
  13. Yiming Chen, "Gaussian Convexity Principles for Sharp Moderate Deviations of Gaussian Maxima and Critical SK Free Energy Variance", arXiv:2607.21392 (2026), https://arxiv.org/abs/2607.21392
  14. Hang Du and Brice Huang, "Overlap distribution of the critical Sherrington-Kirkpatrick model", arXiv:2608.08752 (2026), https://arxiv.org/abs/2608.08752