Large deviations for the maximum of the generalized TAP free energy

Technical Summary: Large Deviations for the Maximum of the Generalized TAP Free Energy

Problem Statement
This paper investigates the large deviations of the maximum of the generalized TAP (Thouless-Anderson-Palmer) free energy for the Ising mixed $p$-spin model. The TAP free energy, introduced by Chen, Panchenko, and Subag, represents the free-energy cost of fixing the barycenter of many replicas at a specific magnetization vector $m$. While the Parisi formula for the equilibrium free energy is rigorously established, the probabilistic interpretation of "supersymmetric" (SUSY) formulas for the complexity of TAP states has remained ambiguous.

Physicists have long conjectured that SUSY calculations yield the annealed complexity (the exponential growth rate of the number of critical points). However, the paper highlights a controversy: dropping the absolute value of the Hessian determinant in SUSY calculations is only valid when the Hessian maintains a fixed signature. In general, non-supersymmetric branches may dominate, rendering the SUSY formula incorrect for the ordinary annealed count. The central question addressed is: What does the SUSY formula actually count? The paper posits that it corresponds not to the ordinary annealed complexity, but to the large-deviation exponent for the existence of TAP states at a given energy level.

Methodology
The proof strategy combines probabilistic large deviation theory with rigorous spin-glass techniques, specifically adapting methods from Huang and Sellke [15] and utilizing a new Guerra-type interpolation.

1. Upper Bound:
   The upper bound on the probability of existence is derived by bounding the exponential moments of the maximum TAP free energy. The author introduces a new Guerra-type interpolation that compares the TAP free energy to an additive Ruelle probability cascade (RPC). This involves:

   • Constructing a multiscale cavity field and an Onsager field on a tree structure defined by the RPC.
   • Using Slepian's inequality to compare the Hamiltonian augmented by the Onsager field against the cavity field.
   • Applying Fenchel duality and maximal estimates along the Parisi flow to bound the supremum.
   • This approach yields a bound on the exponential moment $\mathbb{E}[\exp(\theta \max \text{FTAP})]$, which translates via Markov's inequality to an upper bound on the existence probability.

2. Lower Bound:
   The lower bound follows the strategy of Huang and Sellke [15] but is adapted to the Ising TAP landscape. The core argument involves:

   • Annealed Complexity Calculation: Using the Kac-Rice formula to compute the expected number of "supersymmetric" (SUSY) critical points at a specific energy level. These are points where the gradient vanishes, the self-overlap matches a specific parameter $q$, and the Hessian is negative definite.
   • Isolation and Spherical Codes: The paper demonstrates that SUSY critical points with high energy are "isolated." Specifically, distinct SUSY states cannot have overlaps in the interval $[\delta, q-\delta]$ (due to the strict Parisi obstacle gap) nor in $[q-\delta, q]$ (due to the strict Plefka condition ensuring a negative Hessian).
   • Consequently, distinct retained states form a spherical code with pairwise overlaps less than $\delta$.
   • A one-sided spherical code bound is used to convert the annealed count (first moment) into a lower bound on the probability of existence. This step effectively turns the "annealed count" into a "quenched" existence probability because the states are sufficiently sparse and non-interacting in the relevant regime.

3. Technical Assumptions:
   The results rely on a strict Plefka condition (strict stability of the Hessian) and the assumption that the energy level $f$ is "regular" (the Legendre transform defining the rate function is differentiable). The paper notes that these are likely technical conditions that can be removed in future work.

Key Contributions and Results

• Identification of the SUSY Formula: The primary result is the identification of the supersymmetric formula proposed by physicists. The paper proves that the Legendre transform in the bottom-atom mass (the SUSY complexity) corresponds to the large-deviation rate function for the existence of TAP maxima, rather than the ordinary annealed complexity.

  • The rate function is given by $\Sigma(f) = \inf_{\theta \ge 0} [\bar{\Lambda}(\theta) - \theta f]$, where $\bar{\Lambda}(\theta)$ is the constrained Parisi value.
  • This resolves the discrepancy between SUSY formulas and ordinary annealed counts: the SUSY formula counts isolated local maxima, which have a specific probability of existence, whereas the ordinary annealed count includes contributions from clusters of states that may not exist simultaneously.

• Large Deviation Principle for TAP Existence:
  Theorem 1 establishes that for regular levels $f \in (f_{eq}, f_1)$, the probability of the existence of a TAP state at level $f$ satisfies:
  $$\lim_{N \to \infty} \frac{1}{N} \log \mathbb{P}(E_{N,\epsilon}(f)) = \Sigma(f)$$
  where $E_{N,\epsilon}(f)$ is the event that a critical point exists with energy close to $f$.

• Connection to Free Energy Large Deviations:
  The paper shows that the rate function $\Sigma(f)$ is identical to the upper-tail rate function for the ordinary free energy (as identified by Talagrand). This is natural because an upper deviation in the free energy is realized by the existence of a TAP state at a correspondingly high level.

• Constructive Proof of the Parisi Formula:
  The author argues that the same iterative argument, applied in bands above TAP ancestors, could provide a constructive proof of the Parisi formula for the Ising model. By constructing exponentially branching families of near-optimal points at successive contact levels and organizing them into an ultrametric tree, one could derive the lower bound of the Parisi formula. This extends the work of Huang and Sellke [15] from spherical models to the Ising model, provided a technical strict stability assumption holds.

Significance
The paper provides a rigorous probabilistic interpretation for supersymmetric calculations in spin-glass theory, clarifying that they describe the large deviations of the maximum of the TAP free energy. It bridges the gap between non-rigorous physics heuristics and rigorous probability theory. Furthermore, it offers a pathway toward a constructive proof of the Parisi formula for the Ising model by leveraging the geometry of the TAP landscape and the properties of supersymmetric critical points. The work validates the use of SUSY manipulations for counting isolated local maxima while correcting their interpretation regarding the ordinary annealed complexity.