What carries upper large deviations of the free energy in mixed p-spin Ising spin glasses?

The TAP maximum carries the free energy at every rate

The paper's first result answers the question in its title at the level of the free energy itself. Let M_N be the maximum of the generalized TAP free energy F_TAP of Chen, Panchenko and Subag over the cube [-1,1]^N, and let log Z_N be the log partition function of the mixed p-spin Ising spin glass. Under condition (R), that the radius of convergence of the covariance series exceeds one, the paper proves that M_N is attained almost surely and that for every t > 0 the probability that |M_N - log Z_N| exceeds tN decays at the maximal exponential rate: the limit of N^{-1} log P(|M_N - log Z_N| > tN) equals -1.

That statement is stronger than the earlier band theorem of Chen, Panchenko and Subag, which gave the same comparison only outside an event of probability e^{-c(t)N} for some positive c(t). The paper needs the comparison to hold at every exponential rate, because that is what lets large-deviation statements pass in both directions between M_N and log Z_N. The proof combines the band theorem with Ramsey's theorem, which finds, for every realization of the disorder, an overlap level at which the Gibbs measure puts mass bounded below independently of N into the band around a barycenter of replicas.

A second part of the same theorem transfers the large deviation principle itself: N^{-1} M_N satisfies a large deviation principle at speed N with a good rate function if and only if N^{-1} log Z_N does, and then the two rate functions coincide. The theorem does not yet say where the maximizers of F_TAP sit, at which self-overlap, or whether they are critical points of a TAP function; the barycenter used in the proof sits at an overlap level that depends on the realization. The remaining results answer those questions.

Critical points exist without the strict Plefka condition

The paper's second main result locates the deviating states. For ξ in the class C used by Chen, Panchenko and Subag, every maximizer x* of F_TAP over the cube lies in the interior (-1,1)^N, and if ζ* is the unique minimizer of the order-parameter functional at x*, then F_{ζ*} is continuously differentiable there with vanishing gradient and F_{ζ*}(x*) = F_TAP(x*) = M_N. So a maximizer is itself a witness: a pair (ζ, x) with ∇F_ζ(x) = 0 whose TAP value sits at the same level.

The paper then shows that such generalized TAP critical points exist at the free-energy rate at every upper level f in the interval (f_eq, f_1), where f_eq is the limiting free energy and f_1 is the right endpoint of the interval on which the deviation exponent is finite. This is where the paper removes a hypothesis used by Boursier: her theorem required the first positive contact to be strictly Plefka stable, and the paper shows that for convex mixtures in C the strict Plefka condition and the regularity assumption can both be dropped. Every level in (f_eq, f_1) is regular, with a unique minimizer θ_f in (0,1), and marginal first contacts are allowed.

The paper is explicit that existence here is not elementary. Through the transfer theorem, the lower bound on the probability of a near-maximizer at level f implies the lower half of the fractional-moment formula on (θ*, 1), and conversely, given the upper half, the existence statement is equivalent to that lower bound. For convex ξ the upper half is Guerra's bound; for even ξ the lower half is due to Talagrand; for convex ξ that is not even the paper derives it from the Kac-Rice construction behind its stability result. The paper also notes that the critical points it produces are critical points of F_{ζ*} for an order parameter that depends on the point; they are not shown to be isolated, strictly stable, or critical points of the fixed function F_{ζ_f}.

Near-maximizers localize on the obstacle's contact set

To describe the geometry of the rare states, the paper reads off the first variation of the Parisi functional at the constrained minimizer ζ_f. The obstacle O_ζ is defined by integrating ξ''(s)(Γ_ζ(s) - s) from t to 1, where Γ_ζ(t) is the second moment of ∂_x Φ_ζ(t, X_t) along the Auffinger-Chen process. The first variation of the Parisi functional in the direction of a measure μ is the integral of O_ζ against (μ - ζ), so the minimizers of the obstacle are the cheapest places to move mass to. The contact set C_f is the argmin of O over (0,1]; it is a nonempty compact subset of (0,1) containing the support of ζ_f away from the endpoints, and O(0) is strictly larger than the minimum.

The paper's third theorem states that, outside an event of smaller exponential order, near-maximizers at level f lie near this contact set. The same theorem gives the field law: the law of ∂_x Φ_{ζ_f}(q, X_q) along the Auffinger-Chen process, whose second moment Γ_{ζ_f}(q) equals q at every contact because O' vanishes there. This is the Auffinger-Chen field law referred to in the abstract.

The contact set is also where the Plefka factor is defined. Writing a_Q = ξ''(Q), U_Q = ∂²_x Φ_{ζ_f}(Q, X_Q), χ_Q = E U_Q and Δ(Q) = 1 - a_Q E U_Q², the paper shows that at a contact Q in C_f the obstacle is twice differentiable with O''(Q) = (1/2) a_Q Δ(Q) ≥ 0, because Q is an interior minimum. So the Plefka inequality Δ ≥ 0 holds on the contact set. A contact is strictly Plefka stable when Δ(Q) > 0 and marginal when Δ(Q) = 0, meaning the obstacle is flat to second order there. The distinction matters because the strict condition is exactly what the paper is able to remove for convex mixtures.

The Hessian bulk follows a reflected Pastur law

The fourth part of the third theorem describes the curvature of the TAP function at these rare states. For a ≥ 0 and a probability measure ρ on [0,1), the reflected Pastur law ν[a, ρ] is the unique probability measure on the real line whose Cauchy transform g satisfies g(z) = ∫ ρ(dD) / (z + D - a g(z)) for every z in the upper half plane. The paper shows that this law arises as the limit of the empirical spectral distribution of √a W_N D_N, where W_N is a GOE matrix with off-diagonal variance 1/N and the D_N are diagonal matrices whose empirical spectral distributions converge to ρ.

At a contact Q the paper defines ρ_Q as the law of U_Q^{-1} + a_Q χ_Q and ν_Q as the reflected Pastur law ν[a_Q, ρ_Q], with λ_+(Q) the supremum of its support. The Hessian bulk at the rare TAP states is near this reflected Pastur law, and its edge λ_+(Q) is nonpositive. The edge vanishes exactly at contacts where the obstacle is flat to second order, that is, at the marginal contacts where Δ(Q) = 0. This is the precise sense in which marginal stability and the location of the spectral edge are the same phenomenon in this model.

The paper also records a strict Hessian bound around a conditioned critical point: for a law π and a level h, admissible data of small width have one-sided Hessians bounded away from zero with probability tending to one, provided a certain Schur complement S(π, h) is negative. At the data of a contact the Schur complement has an explicit value, S(π_0, h_0) = q a θ Δ_0 - q Δ_0² / υ_0, which is strictly negative when Δ_0 > 0. When Δ_0 = 0 the paper perturbs the law by a small η and shows that S(π_η, h_η) ≤ -η for every L ≥ L_S = (C_S + 1)/|A_2|, while Δ_η = a η E_0 R² > 0. That is how the marginal case is handled rather than assumed away.

A fixed stability margin costs exponential rate

The paper's fifth result quantifies the price of stability. Where every contact is marginal, in the sense that the obstacle is flat to second order at each point of the contact set, requiring a fixed stability margin costs rate: the probability of finding a near-maximizer with that margin is exponentially smaller than the probability of finding one at the level itself. The abstract states this as: where every contact is of this kind, a fixed stability margin costs rate.

This is the result that connects the local geometry of the contact set to the exponential cost of the event. It also explains why the strict Plefka condition was a natural hypothesis to try to remove: at marginal first contacts the strict condition is false, and the paper's transfer argument for M_N needs no Hessian estimate at all, only a deterministic comparison of the perturbed and unperturbed TAP functions together with an explicit Gaussian tail bound for the perturbing field. The paper notes that Boursier used the same perturbation but transferred retained critical points under the strict Plefka condition, which fails at marginal first contacts.

The perturbation used is a small even-degree term: ξ_γ(t) = ξ(t) + γ² t^l and H^γ_N = H_N + γ H^{(l)}_N, with d_γ = l γ². The comparison of covariances gives |F^γ_TAP(x) - F_TAP(x) - γ H^{(l)}_N(x)| ≤ d_γ N for every realization and every x, hence |M^γ_N - M_N| ≤ γ S_N + d_γ N, where S_N is the supremum of the perturbing field. The order of limits is fixed: fix f and ε, choose γ, let N tend to infinity, and only then let γ decrease to zero. No estimate needs to be uniform in γ.

Where the results apply and where they stop

The hypotheses are stated level by level, and they matter for reading the conclusions. Theorem 1, the transfer between M_N and log Z_N, assumes only (R), that the radius of convergence of ξ(t) = Σ_{p≥2} β_p² t^p exceeds one. Theorem 3(i)-(ii) and the corollary on the equilibrium level assume (H), that ξ satisfies (R) and is convex on [-1,1]. Theorem 2 assumes ξ in the class C of Chen, Panchenko and Subag, which contains every polynomial and forces radius of convergence at least two. Theorem 3(iii)-(iv), the corollary on the Hessian, and Theorem 5 assume both (H) and ξ in C.

The class C enters only through cited facts about the Parisi equation and the TAP correction; part (b) of Theorem 2 and the uniqueness of the constrained minimizers need only (R). The paper also notes that every even ξ satisfying (R) is convex on [-1,1], since ξ'' is then a series of even powers with nonnegative coefficients, so (H) is the standing assumption of Boursier. Odd degrees are allowed under (H), but a convex ξ with an odd degree has at least two active degrees, because a single odd monomial is not convex on [-1,1]. The convex covariances with one active degree are the Sherrington-Kirkpatrick model ξ(t) = β² t² and the pure even p-spin models.

The results are stated for levels f in the open interval (f_eq, f_1). The interval is nonempty exactly when θ* = ζ_P({0}) is less than one, and it is empty when the Parisi measure is δ_0. For f ≥ f_1 the paper has only the upper bound P(N^{-1} log Z_N ≥ f) ≤ e^{N Σ(f)}, which holds for every N by Markov's inequality and E Z_N = e^{N ξ(1)/2}; at those levels the infimum defining Σ(f) is attained at θ = 1 and Σ(f) = ξ(1)/2 - f. The paper treats f in (f_eq, f_1) throughout.

The paper also flags what its critical points are not. The points of Theorem 2(a) are critical points of F_{ζ*} for an order parameter ζ* that depends on the point; they are not shown to be isolated, to be strictly stable, or to be critical points of the fixed function F_{ζ_f}. For convex ξ with at least two active degrees, Theorem 5(a) supplies critical points of F_{ζ_f} that are locally strongly concave, hence isolated, at rates arbitrarily close to Σ(f). The paper also notes that the events quantifying over all pairs (ζ, x) need not be measurable, so P stands for the inner probability in lower bounds and the outer probability in upper bounds, and every stated equality of limits holds for both.

The marginal contact is the case that changes the picture

The most decision-relevant distinction in this paper is between strictly Plefka stable contacts and marginal ones. Earlier work by Boursier proved the corresponding existence and rate statements under the strict Plefka condition, which requires Δ(Q) > 0 at the first positive contact, and she expected the strictness to be removable. This paper shows that for convex mixtures in C it is removable: every level in (f_eq, f_1) is regular, and marginal first contacts are allowed. The paper identifies two settings where marginal first contacts actually occur: the Sherrington-Kirkpatrick model at low temperature at all levels close to f_eq, and an interval of levels in a family of near-critical 2 + 4 mixtures.

The second distinction is between the two objects that the paper proves to have the same large deviations. M_N is a maximum over the cube of an infimum over order parameters, while log Z_N is a log partition function; the transfer theorem makes their large deviation rate functions coincide, and the transfer runs in both directions. The paper is careful that this does not by itself locate the maximizers, which is why the localization, field-law, and Hessian results are separate theorems rather than corollaries of the transfer.

The third distinction is between the level f and the stability margin. The paper's stability result says that requiring a fixed margin at marginal contacts costs exponential rate, so the states that carry the deviation are not the stable ones but the marginal ones. For a reader deciding which regime to study, the paper's own scenario split is therefore: convex mixtures in C at levels in (f_eq, f_1), where marginal contacts are permitted and the rate is Σ(f); levels at or above f_1, where only the Markov upper bound is available; and the strict Plefka case, which the paper subsumes rather than replaces. The paper's contribution is to show that the marginal case is not a degenerate exception but the case that the earlier strict condition was excluding.

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