What is the fifth-order law for upper deviations of the Sherrington-Kirkpatrick free energy?

The constant that earlier bounds left open

Replica theory predicted the shape of the upper tail long before it was proved. Kondor found a term 9n⁵/5120 in the replica free energy of a truncated model near the critical temperature, and Parisi and Rizzo extended the computation to every temperature below the critical one, predicting that the constrained Parisi minimum p(θ) exceeds the equilibrium free energy f_eq by c_β θ⁵ with c_β = (9/640) β³ ρ_β(0)⁻³. Here ρ_β(0) is the density at zero of the Parisi measure, the reciprocal of the slope q̇(0) at x = 0 of Parisi's order parameter.

Aronow and Lopatto recently proved that p(θ) − f_eq is bounded above and below by constant multiples of θ⁵ for the Sherrington-Kirkpatrick model at every β>1, and that at finite N the probability that F_{N,β} exceeds N(f_eq + log 2 + x) lies between e^{−CNx^{6/5}} and e^{−C′Nx^{6/5}} for AN^{−2/3} ≤ x ≤ x₀. Their constants were not specified. This paper supplies them, and the value it finds is the leading term of the Parisi-Rizzo prediction.

Why a fifth-order excess becomes a six-fifths exponent

Upper deviations are governed by fractional moments. Markov's inequality gives P(N⁻¹ log Z_N ≥ f) ≤ e^{−Nθf} E Z_N^θ for 0<θ<1, and Talagrand related the growth rate of E Z_N^θ to the Parisi problem constrained to measures carrying an atom of mass at least θ at zero. Writing p(θ) for that constrained minimum, the fractional-moment formula reads lim_N N⁻¹ log E Z_N^θ = Λ(θ) := θ p(θ), and for the Sherrington-Kirkpatrick model it is the one-level case of a theorem of Contucci and Mingione.

The rate near f_eq is therefore decided by how fast p(θ) decreases to f_eq. If p(θ) − f_eq is of order θ⁵ as θ↓0, the rate at f_eq + δ is of order δ^{6/5}. That is the bridge between the variational statement about constrained measures and the probabilistic statement about the upper tail, and it is why the paper's central quantity is a fifth-order excess rather than a tail exponent directly.

The constrained minimizer's first contact is marginal

The constant is not an abstract fitting parameter; it comes from the shape of the constrained minimizer ζ_θ near the bottom of its support, at scale θ. For small θ the first positive contact is marginal, sits at 3θ/(2ρ_β(0)) (1+o(1)), and carries an atom of mass θ/2 (1+o(1)). Below that contact, a function Γ_θ that equals the identity at the contact points differs from the identity, to leading order, by a universal cubic.

This is a local statement at the origin. The paper shows the same plateau calculus applies to every mixed p-spin model without a cubic term whose Parisi measure accumulates at zero, so the mechanism is not special to the Sherrington-Kirkpatrick covariance. The constant c_β = (9/640) β³ ρ_β(0)⁻³ is the value produced by minimizing over the height of the first plateau, and the paper obtains it from an envelope identity and the universal cubic rather than from the trial-measure expansion used in the replica literature.

Two other perturbations of the bottom of the measure

The same local calculus governs two further perturbations. A small external field h opens a gap of order |h|^{2/3} and adds a term of order |h|^{10/3} to the free energy; the paper records q_min(βH) = (3T q̇(0)/2)^{1/3} |H|^{2/3}(1+o(1)) and f(H) − f(0) = H²/2 + (3/20)(9/(4T q̇(0)))^{1/3} |H|^{10/3}(1+o(1)), with T = 1/β. The quadratic term H²/2 is Toulouse's identity F″(0) = 1/β.

An atom of the Parisi measure at zero, which every pure p-spin model with p≥3 has, makes the rate function leave zero linearly, with slope equal to that atom. These are different perturbations of the same object, and they should not be read as one result: the field case is a gap of order |h|^{2/3} in the measure and an |h|^{10/3} free-energy correction, while the atom case changes the local behaviour of the rate function itself.

Near criticality the constant tends to Kondor's value

As β↓1 the constant c_β tends to Kondor's value 9/5120, so the older near-critical computation appears as a limit of the all-temperature result rather than as a separate regime. The paper also shows that the marginal regime ends at a de Almeida-Thouless threshold in the constraint: for 1<β²<1+ε*, the threshold is θ_c = (2/3)(β²−1) + O((β²−1)²), whose leading term matches the replica-symmetric line reported by Parisi and Rizzo.

Above its first contact the constrained measure agrees with the Parisi measure up to O(θ⁵). This is the constrained-Parisi form of the prediction that q(x) = q_free(x) + O(n⁵) above the first plateau, and it is proved here near criticality rather than at every temperature below the critical one.

What the two limits do and do not cover

All limits are taken at fixed β, with N→∞ before δ↓0. The paper makes no statement about deviations that shrink with N, the o(1) terms carry no rate, and nothing is uniform in β. The statement at every β>1 uses Lopatto's theorem that zero is an accumulation point of the support of the Parisi measure; the local results assume this condition (H), and when ξ′′′(0)=0 it implies the stronger condition that [0,κ] lies in the support for some κ>0.

Condition (H) fails whenever ξ″(0)<1, for instance for every pure p-spin model with p≥3, and beyond the Sherrington-Kirkpatrick model and the near-critical covariances the paper does not know a proof of (H) for genuinely mixed covariances. The paper proves no correction term and nothing about the fluctuations of order N^{−5/6} that Parisi and Rizzo relate to their exponent by a matching argument they describe as not rigorous.

What the identified constant changes for a reader

The decision-relevant shift is that the upper tail of N⁻¹ log Z_N is no longer known only up to unspecified β-dependent constants. For the Sherrington-Kirkpatrick model at every β>1, the rate at f_eq + δ is (5/6)(6c_β)^{−1/5} δ^{6/5}(1+o(1)) with c_β = (9/640) β³ ρ_β(0)⁻³, and the same statement holds for F_{N,β} after N⁻¹ log Z_N is replaced by N⁻¹F_{N,β} − log 2.

The quantity to watch is ρ_β(0), the density at zero of the Parisi measure, equal to β⁵W_*²/2 with W_* the fourth spatial derivative at the origin of the solution of the Parisi equation for ζ_*. It is this density, not the exponent 6/5, that carries the β-dependence of the constant, and it is the first positive contact at 3θ/(2ρ_β(0)) together with its atom of mass θ/2 that produces it. Readers comparing this result with earlier work should compare constants at fixed β, not exponents, and should keep the two limits in the stated order.

Sources used in this answer