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Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

2026 · arxiv_cs

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NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE IN THE SPHERICAL PURE p-SPIN MODEL

arXiv:2608.14369v1 [math.PR] 14 Aug 2026

TAEGYUN KIM Abstract. We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure p-spin glasses with overlap q. For every p ≥ 3 and p 0 < β ≤ βsh (p), we rule out shattering whenever q ≤ 2−1/2 or q > (p − 2)/(p − 1). The proof combines a deterministic N + 1 bound for disjoint bands in the first range with a general-p sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder’s inequality give an additional q-dependent obstruction; in particular, they rule out every √ fixed overlap for 0 < β ≤ log 2. For p = 3, the first two ranges already exhaust every fixed q ∈ (0, 1), so the landscape is not shattered at any T ≥ Tsh . For p p ≥ 4, the cases not covered by our criteria are confined to √ 2−1/2 < q ≤ (p − 2)/(p − 1) and log 2 < β ≤ βsh (p). In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

1. Introduction The spherical pure p-spin model is a canonical mean-field model in which the geometry of a random high-dimensional landscape can be related to the thermodynamics and dynamics of a glassy system. Its statics and dynamics were analyzed in the foundational works [7, 6]. Rigorous critical-point complexity and the low-temperature decomposition of the Gibbs measure into bands around deep critical points were subsequently developed in [1, 14]; see also [15] for the associated band free-energy landscape and generalized TAP picture. Ben Arous and Jagannath introduced in [4] a precise freeenergy notion of shattering: the Gibbs free energy is asymptotically carried by exponentially many pairwise disjoint bands around critical points, while every individual band has exponentially small Gibbs mass. Their Conjecture 1 predicts shattering for Ts ≤ T ≤ Tsh and non-shattering for T > Tsh . Thus its printed formulation assigns the endpoint T = Tsh to the shattered side. We prove a partial non-shattering theorem for every spherical pure p-spin model on the high-temperature side and, for p = 3, a complete non-shattering Date: August 13, 2026. 2020 Mathematics Subject Classification. 60G15, 60K35, 82B44, 82D30. Key words and phrases. spherical spin glass, pure p-spin model, shattering, Kac–Rice formula, complexity, dynamical transition. 1

TAEGYUN KIM

theorem for all fixed overlaps and every T ≥ Tsh . We work throughout with the exact fixed-overlap, critical-point-band definition in [4, Definition √ 2.3]. The argument also gives an all-overlap result for every p ≥ 3 when β ≤ log 2, and identifies the smaller region that remains unresolved for p ≥ 4. More recent works use stronger or different notions of shattering; see [9, 2]. We make no assertion about those other notions. We write Ts for the static replica-symmetry-breaking temperature and Tsh for the dynamical, or shattering, temperature, following the convention of [4]. Fix an integer p ≥ 3. Let √ x·y SN = S N −1 ( N ) ⊂ RN , R(x, y) = , N and let HN,p be the centered Gaussian field with covariance EHN,p (x)HN,p (y) = N R(x, y)p .

(1.1)

When p is fixed, we write HN = HN,p . Set βsh (p)2 =

(p − 1)p−1 , p(p − 2)p−2

Tsh (p) = βsh (p)−1 ,

s∗ (p) =

For p = 3, the formula in [4, Section 2] gives √ 3 2 Tsh = Tsh (3) = , βsh = βsh (3) = √ , 2 3 Our main result is the following.

p−2 . p−1

(1.2)

1 s∗ (3) = . 2

Theorem 1.1 (Main theorem). Fix p ≥ 3, 0 < β ≤ βsh (p), E ∈ R, and r ≥ 0. The spherical pure p-spin free-energy landscape is not (E, q, r)shattered whenever 1 q ∈ 0, √ ∪ 2

p−2 ,1 . p−1

(1.3)

Equivalently, the inverse-temperature assumption is T ≥ Tsh (p). For q > 2−1/2 , the same conclusion also holds whenever β 2 < −2RLP (2q 2 − 1) − log(1 − q 2 ),

(1.4)

where the spherical-code rate RLP is defined in (3.4) below. In particular, when p = 3, the two intervals in (1.3) exhaust q ∈ (0, 1). Hence the spherical pure 3-spin landscape is not (E, q, r)-shattered for any fixed E ∈ R, q ∈ (0, 1), and r ≥ 0, throughout T ≥ Tsh . Consequently, for every p ≥ 3, the landscape is not (E, q, r)-shattered for any fixed E ∈ R, q ∈ (0, 1), and r ≥ 0, whenever 0<β≤

log 2,

equivalently

T ≥ (log 2)−1/2 .

(1.5)

Corollary 1.2. For p = 3, the non-shattering assertion in Conjecture 1 of [4] for T > Tsh holds and extends to T = Tsh . Consequently, both the inclusion of the endpoint in the shattered regime of Conjecture 1 and the endpoint assertion of [4, Theorem 8.4] are false as printed.

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

For p = 3, the specialization of Theorem 1.1 proves the conjectured nonshattering statement for T > Tsh and extends it to equality. The new issue is equality at the endpoint, and the discrepancy there is already visible in the published proof. Although [4, Theorem 8.4] includes T = Tsh , its joint proof with Theorem 2.5 begins under the strict assumption βsh < β < Ts−1 ; see the proof of Theorems 2.5 and 8.4 in [4, Section 5]. The discussion specialized to p = 3 again uses β > βsh to place the relevant overlap strictly above the geometric threshold; see [4, Section 8]. The proof is organized around a variational statement valid for general p. For a critical point x, write u = −HN (x)/N , set s = q 2 , and let Φp,β be the marked exponent defined in (4.6). We prove that, for every 0 < β ≤ βsh (p), √ β2 sgn sup Φp,β (u, s) − 2 u∈R

   1,

0,   −1,

0 ≤ s < s∗ (p), s = s∗ (p), s∗ (p) < s < 1.

(1.6)

When p = 3, the transition point is s∗ = 1/2. Pairwise disjointness gives at most N + 1 centers for s ≤ 1/2, whereas the marked Kac–Rice bound gives a strict free-energy loss for s > 1/2. These alternatives cover every fixed overlap and every 0 < β ≤ βsh . For p ≥ 4, the transition point satisfies s∗ (p) > 1/2. The interval 1/2 < s ≤ s∗ (p) is therefore controlled by neither of these two arguments: the deterministic bound no longer applies, while the marked exponent has no strict negative gap. A third argument, based on the asymptotic linearprogramming bound for spherical codes and Hölder’s inequality, supplies a further q-dependent obstruction and settles this entire interval when √ √ β≤ log 2. Thus the cases not covered by our criteria are confined to log 2 < β ≤ βsh (p). The random-landscape input uses only one-point Kac–Rice analysis. The extra Gibbs mark is handled by Gaussian regression. Conditional on the value and gradient at a prospective critical point, the Hamiltonian at a point on a latitude has the standard conditional mean and variance. Its covariance with the conditional Hessian has rank at most one. Exponential tilting therefore changes the conditional Hessian only by a bounded rank-one perturbation. A cofactor expansion, combined with the expected absolute-determinant asymptotics in [3, Corollary 1.3], shows that this perturbation does not alter the exponential determinant rate. This yields the marked exponent in (4.6). The rest of the paper is organized as follows. We first set notation and recall the shattering definition, then prove the deterministic small-overlap obstruction and a spherical-code entropy bound. We next prove the marked Kac–Rice bound, establish the general-p sign law, and combine it with the small-overlap geometry to prove Theorem 1.1. We then specialize to p = 3 to record the complete all-overlap conclusion and a sharper fourth-order endpoint tangency. The final section discusses the endpoint claim and the relation with other notions of shattering.

TAEGYUN KIM

2. Model, free energy, and shattering Let µN be the normalized uniform measure on SN . For a Borel set A ⊆ SN , define Z 1 ZN,β (A) = e−βHN (σ) µN ( dσ), FN,β (A) = log ZN,β (A). (2.1) N A We abbreviate ZN,β = ZN,β (SN ),

FN (β) = FN,β (SN ).

For x ∈ SN , q ∈ (−1, 1), and η > 0, let

B(x, q, η) = σ ∈ SN : |R(x, σ) − q| ≤ η .

(2.2)

For fixed x ∈ SN , the overlap of a µN -uniform point with x has density Γ(N/2) (1 − t2 )(N −3)/2 , −1 < t < 1. (2.3) νN (t) = √ π Γ((N − 1)/2) For an interval I ⊂ R, write HN (x) CN (I) = x ∈ SN : ∇HN (x) = 0, ∈I . (2.4) N We record the definition from [4] in the form used below. Definition 2.1 (Ben Arous–Jagannath shattering). Fix T > 0, E ∈ R, q ∈ (0, 1), and r ≥ 0, and set β = T −1 . The free-energy landscape is (E, q, r)shattered if there exist constants c, c′ > 0, positive sequences εN , ηN , δN → 0, and random sets AN ⊆ CN ([−E − εN , −E + εN ]) such that, with probability tending to one, (1) N −1 log|AN | ≥ c; (2) the bands B(x, q, ηN ), x ∈ AN , are pairwise disjoint, and R(x, y) < r for distinct x, y ∈ AN ; (3) every band is exponentially subdominant: FN (β) − FN,β (B(x, q, ηN )) > c′ ; (4) the union has the full limiting free energy: 

FN (β) − FN,β 

 [

B(x, q, ηN ) − → 0.

x∈AN

The three vanishing sequences are deterministic and positive, the finite random sets AN are measurable, conditions (1)–(3) hold jointly with probability tending to one, and condition (4) is a convergence-in-probability statement. Remark 2.2 (Fixed overlap parameter). As in the published definition, E, q, and r are fixed independently of N . In particular, Theorem 1.1 does not address modified notions in which q = qN . Sequences approaching either p −1/2 2 or (p − 2)/(p − 1) lie outside the definition considered here; for p = 3, the two boundary values coincide.

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

Throughout the high-temperature range of Theorem 1.1, the limiting free energy is replica symmetric. Lemma 2.3. Fix p ≥ 3 and 0 < β ≤ βsh (p). Then P

FN (β) − →

β2 . 2

(2.5)

Proof. For a spherical model with covariance function ξ(t) = tp , the standard replica-symmetry criterion states that the limiting free energy equals β 2 ξ(1)/2 provided gβ (t) = β 2 tp + log(1 − t) + t ≤ 0, 0 ≤ t < 1; . The spherical Parisi formula was proved for even mixtures in [16] and for general mixtures, including odd p, in [5]; the above criterion in the present pure setting is recorded in [4, Equation (3.8)]. At β = βsh (p), gβ′ (t) =

t 2 p−2 β pt (1 − t) − 1 ≤ 0. 1−t

Indeed, tp−2 (1 − t) is maximized at t = (p − 2)/(p − 1), where the parenthesized expression vanishes. Since gβ (0) = 0, this gives gβ (t) ≤ 0 at the endpoint. The same inequality holds for every smaller β. The criterion and concentration of the spherical free energy yield (2.5). □ 3. A deterministic obstruction for small band overlap For x ∈ SN and q ∈ (−1, 1), define the exact latitude L(x, q) = {σ ∈ SN : R(x, σ) = q}.

(3.1)

Clearly L(x, q) ⊂ B(x, q, η) for every η > 0. Lemma 3.1 (Intersection criterion for exact latitudes). Assume N ≥ 3. Let x, y ∈ SN be distinct and set ρ = R(x, y). For q ∈ (0, 1), L(x, q) ∩ L(y, q) ̸= ∅

⇐⇒

ρ ≥ 2q 2 − 1.

(3.2)

Consequently, if B(x, q, η) ∩ B(y, q, η) = ∅ for some η > 0, then R(x, y) < 2q 2 − 1.

(3.3)

Proof. A point σ ∈ SN with R(x, σ) = R(y, σ) = q exists if and only if the Gram matrix 



1 ρ q G = ρ 1 q  q q 1 is positive semidefinite. Indeed, necessity is immediate; conversely, a positive semidefinite 3 × 3 Gram matrix has a realization in RN because N ≥ 3,

TAEGYUN KIM

√ √ and an orthogonal map can match its first two vectors to x/ N and y/ N . Since x ̸= y, we have ρ < 1, and det G = (1 − ρ)(1 + ρ − 2q 2 ). The principal 2 × 2 minors are nonnegative, so G ⪰ 0 is equivalent to 1 + ρ − 2q 2 ≥ 0, proving (3.2). The last assertion follows from L(x, q) ⊂ B(x, q, η). □ We use the following classical simplex bound for spherical codes, due to Rankin [13, Theorem 1]; a proof is included for convenience. Lemma 3.2. Let v1 , . . . , vM ∈ RN be unit vectors satisfying vi · vj < 0

(i ̸= j).

Then M ≤ N + 1. Proof. Suppose M > N + 1. The points v1 , . . . , vM are affinely dependent, so there are real numbers ai , not all zero, such that M X

M X

ai vi = 0,

i=1

ai = 0.

i=1

There are both positive and negative coefficients. After splitting the indices into P = {i : ai > 0} and M− = {j : aj < 0}, and normalizing, we obtain positive coefficients αi , βj with each family summing to one and z :=

αi vi =

i∈P

βj vj .

j∈M−

Taking the inner product of the two representations gives ∥z∥2 =

X X

αi βj (vi · vj ) < 0,

i∈P j∈M−

□

a contradiction.

Proposition 3.3 (No positive complexity for q ≤ 2−1/2 ). Assume N ≥ 3, and fix q ∈ (0, 2−1/2 ] and η > 0. If A ⊆ SN and the bands {B(x, q, η) : x ∈ A} are pairwise disjoint, then |A| ≤ N + 1. Hence condition (1) in Definition 2.1 cannot hold. Proof. By Lemma 3.1, distinct centers satisfy R(x, y) < 2q 2 − 1 ≤ 0. √ Applying Lemma 3.2 to the unit vectors x/ N , x ∈ A, gives |A| ≤ N +1.

□

We next quantify what remains possible once the overlap threshold in Proposition 3.3 becomes positive. Let Asph N (a) denote the largest cardinality

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

of a subset of the unit sphere S N −1 (1) whose distinct elements have inner product at most a. Write H(x) = −x log x − (1 − x) log(1 − x),

0 ≤ x ≤ 1,

with the convention 0 log 0 = 0, and, for 0 ≤ a < 1, define 1−t 1+t RLP (a) = , H 2t 1+t

1 − a2 .

(3.4)

The asymptotic linear-programming bound of Kabatiansky and Levenshtein [12, Theorem 4 and Equation (54)], after converting their base-two logarithms to natural logarithms, states that for every fixed a ∈ (0, 1), 1 log Asph N (a) ≤ RLP (a). N →∞ N

lim sup

(3.5)

Lemma 3.4 (A comparison for the spherical-code rate). For 0 ≤ a < 1, 1 RLP (a) ≤ − log(1 − a), 2 with equality if and only if a = 0.

(3.6)

Proof. The assertion is immediate when a = 0. For 0 < a < 1, put √ 1−t x= , y = x. 1+t Then 0 < y < 1 and t=

1 − y2 , 1 + y2

2y , 1 + y2

1+t 1 = . 2t 1 − y2

Consequently, 2y 2 log y − log(1 − y) − log(1 + y), 1 − y2 1 1 − log(1 − a) = − log(1 − y) + log(1 + y 2 ). 2 2 RLP (a) = −

Now −2y log y < 1 − y 2 for 0 < y < 1, because g(y) = 1 − y 2 + 2y log y satisfies g(1) = 0 and g ′ (y) = 2(1 − y + log y) < 0, so g(y) > g(1) = 0. Hence −

2y 2 log y < y. 1 − y2

Moreover, if k(y) = log(1 + y) +

1 log(1 + y 2 ) − y, 2

then k(0) = 0 and y 2 (1 − y) > 0. (1 + y)(1 + y 2 ) Combining the last two inequalities proves (3.6), strictly when a > 0. k ′ (y) =

□

TAEGYUN KIM

Proposition 3.5 (Entropy loss for a union of disjoint bands). Fix q ∈ (2−1/2 , 1), let ηN ↓ 0, and let AN ⊆ SN be any possibly random set such that the bands B(x, q, ηN ), x ∈ AN , are pairwise disjoint. There is a deterministic sequence eN = eN (q, (ηk )k ) → 0, independent of AN and of the disorder, such that 



[ 1 log µN  B(x, q, ηN ) ≤ h(q) + eN , N x∈A

1 h(q) < − log 2, 2

(3.7)

where h(q) = RLP (2q 2 − 1) +

1 log(1 − q 2 ). 2

(3.8)

Proof. Put qN = q −ηN . For all large N , qN > 0, and L(x, qN ) ⊆ B(x, q, ηN ). Applying Lemma 3.1 at qN therefore shows that disjointness implies 2 R(x, y) < 2qN −1

(x ̸= y).

Since qN < q, the normalized centers therefore form a spherical code whose distinct inner products are at most 2q 2 − 1. Also, by (2.3), every band has the same measure and µN (B(x, q, ηN )) ≤ VN (qN ) :=

Z 1

νN (t) dt. qN

It follows that 

µN 

 [

2 B(x, q, ηN ) ≤ Asph N (2q − 1)VN (qN ).

(3.9)

x∈AN

For every fixed sufficiently small δ > 0, the right-hand side is at most 2 Asph N (2q − 1)VN (q − δ)

for all large N . The code bound (3.5) and the standard spherical-cap asymptotic 1 1 log VN (v) = log(1 − v 2 ), N →∞ N 2 lim

0 < v < 1,

therefore give, after δ ↓ 0, the first inequality in (3.7). More explicitly, if bN is the logarithm of the deterministic right-hand side of (3.9), divided by N and minus h(q), then lim supN bN ≤ 0. Thus eN = max{bN , 0} + N −1 tends to zero and is uniform over all choices of AN . Finally, Lemma 3.4 applied to a = 2q 2 − 1 > 0, together with 1 − a = 2(1 − q 2 ), gives 1 1 1 h(q) < − log 2(1 − q 2 ) + log(1 − q 2 ) = − log 2. 2 2 2

□

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

Proposition 3.6 (Spherical-code–Hölder obstruction). Fix p ≥ 3, 0 < β ≤ βsh (p), E ∈ R, r ≥ 0, and q ∈ (2−1/2 , 1). Set 1 cLP (q) = −RLP (2q 2 − 1) − log(1 − q 2 ). (3.10) 2 If β 2 < 2cLP (q), (3.11) then the landscape is not (E, q, r)-shattered. Consequently, for every p ≥ 3, E ∈ R, and r ≥ 0, the landscape is not √ (E, q, r)-shattered for any fixed q ∈ (0, 1) whenever 0 < β ≤ log 2. Proof. Suppose that a shattering family exists and write [ 1 UN = B(x, q, ηN ), mN = log µN (UN ). N x∈A N

Let DN be the event that these bands are pairwise disjoint; by condition (2) in Definition 2.1, P(DN ) → 1. For every λ > 1, Hölder’s inequality gives the pathwise bound 1/λ

ZN,β (UN ) ≤ µN (UN )1−1/λ ZN,λβ , 1 1 mN + FN (λβ). (3.12) FN,β (UN ) ≤ 1 − λ λ Set cq = cLP (q) = −h(q). By (3.11), we may choose λ > 1 such that λβ 2 < 2cq . Choose a fixed c satisfying λβ 2 /2 < c < cq . By Proposition 3.5, for all sufficiently large N , DN implies mN ≤ −c. For every θ > 0, Gaussian integration gives 2 EZN,θ = eN θ /2 . Markov’s inequality, together with Lemma 2.3 at β, therefore gives deterministic sequences aN , bN ↓ 0 and events GN , with P(GN ) → 1, on which

λ2 β 2 β2 + aN , FN (β) ≥ − bN . 2 2 For instance, one may take aN = N −1/2 , for which the Markov failure √ probability is at most e− N ; the sequence bN follows by a diagonal choice from convergence in probability. On DN ∩ GN , (3.12) therefore gives FN (λβ) ≤

FN,β (UN ) ≤ − 1 −

1 λβ 2 aN c+ + . λ 2 λ

The deterministic part of the right-hand side lies below β 2 /2 by the positive amount ! c β2 (λ − 1) − . λ 2 It follows that a fixed positive lower bound on FN (β) − FN,β (UN ) holds with probability tending to one, contradicting condition (4) in Definition 2.1.

TAEGYUN KIM

Finally, Proposition 3.5 gives cLP (q) > 21 log 2 for every q > 2−1/2 . Thus 2 β ≤ log 2 implies the strict criterion (3.11) for every such fixed q, while Proposition 3.3 treats q ≤ 2−1/2 . This range is contained in the assumed dynamical replica-symmetric range: with n = p − 2, the binomial theorem gives n+1 1 n 4 βsh (p)2 = 1+ ≥ > log 2. n+2 n 3 □ 4. A marked Kac–Rice upper bound We next control the total partition-function mass of all bands centered at critical points. Throughout this section, p ∈ {3, 4, . . .} is fixed and EHN (x)HN (y) = N R(x, y)p . Put n = N − 1. We use the normalization in which the independent uppertriangular entries of the n × n GOE matrix Gn satisfy 2 1 (i < j), E(Gn )2ii = . (4.1) E(Gn )2ij = n n Thus its empirical spectral law converges to the semicircle law on [−2, 2]. Define s

αN,p =

p(p − 1)(N − 1) , N

KN (u) = αN,p GN −1 + puIN −1 .

(4.2)

Let µsc be the semicircle law

√ 4 − λ2 µsc ( dλ) = 1{|λ|≤2} dλ, 2π and define its logarithmic potential Z 2

Ω(x) = −2

log|x − λ| µsc ( dλ).

(4.3)

The one-point annealed critical-point complexity is 1 1 u2 +Ω Σp (u) = + log(p − 1) − 2 2 2

p u . p−1

(4.4)

For t ∈ (−1, 1), set vp (t) = 1 − t2p − pt2p−2 (1 − t2 ),

ap (t) = p(p − 1)tp−2 (1 − t2 ),

(4.5)

1 β2 log(1 − t2 ) + vp (t). 2 2

(4.6)

ZN,β (B(x, q, η)).

(4.7)

and, for β > 0, define Φp,β (u, t) = Σp (u) + βtp u + For q ∈ (0, 1) and η > 0, let MN (q, η) =

X x: ∇HN (x)=0

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

The sum is finite almost surely by the standard nondegeneracy of the pure spherical model; this is also part of the Kac–Rice setup in [1, Section 3]. For Borel sets A ⊂ R and I ⊂ (−1, 1), write X

KN (A, I) = E

1{−HN (x)/N ∈A}

x: ∇HN (x)=0

Z {σ:R(x,σ)∈I}

e−βHN (σ) µN ( dσ).

(4.8)

Lemma 4.1 (Finite-dimensional critical-point law). For every x ∈ SN and u ∈ R, in an orthonormal tangent frame,

L ∇2 HN (x) | HN (x) = −N u, ∇HN (x) = 0 = L(KN (u)).

(4.9)

Equivalently, the conditional mean is puIN −1 , and the centered conditional covariance is Cov (∇2 HN )ij , (∇2 HN )kℓ | HN , ∇HN =

p(p − 1) (δik δjℓ + δiℓ δjk ). (4.10) N

Proof. This follows by differentiating the covariance kernel. A direct derivation, including the spherical curvature term, is given in Appendix A; compare [1, Lemma 3.2]. □ Define the expected critical-value density ρN by E

f −

x: ∇HN (x)=0

HN (x) N

f (u)ρN (u) du

(4.11)

for every nonnegative Borel function f . Lemma 4.2 (Exact one-point Kac–Rice density). Let √ 2π N/2 N (N −1)/2 ωN = Vol(S N −1 ( N )) = , Γ(N/2) s

κN,p = ωN (2πp)−(N −1)/2

(4.12)

N . 2π

Then 2

ρN (u) = κN,p e−N u /2 E|det KN (u)|,

(4.13)

1 1 1 log κN,p −→ − log p. N 2 2

(4.14)

and

Proof. The gradient has covariance pIN −1 and is independent of the field value. The one-point Kac–Rice formula, the change of variables HN (x) = −N u, and Lemma 4.1 therefore give (4.13); see also [1, Lemma 3.1 and Section 3]. Stirling’s formula gives (4.14). □

TAEGYUN KIM

(1) LN,p (u) −

1 log(p(p − 1)) + Ω 2

p u p−1

−→ 0.

(4.18)

In particular, the limiting determinant rate is well defined by r 1 p (1) Lp (u) := lim LN,p (u) = lim LN,p (u) = log(p(p − 1)) + Ω u . N →∞ N →∞ 2 p−1 (4.19) Consequently, 1 sup log ρN (u) − Σp (u) −→ 0. (4.20) N |u|≤L Proof. Writing KN (u) = αN,p (GN −1 − zN (u)IN −1 ) ,

zN (u) = −

pu , αN,p

we have zN (u) → − p/(p − 1) u locally uniformly. The expected absolutedeterminant asymptotics for Wigner matrices [3, Corollary 1.3] hold locally uniformly in the spectral parameter, including at the spectral edges. They give (4.17). Moreover, s d (1) GN −1 =

N −2 GN −2 , N −1

(4.21)

so the same result gives (4.18). Finally, (4.13), (4.14), and (4.19) give (4.20), which also recovers the standard complexity formula (4.4) of [1]. □ Fix x ∈ SN , and choose σ ∈ SN with R(x, σ) = t. We use the shorthand Ex,u = {HN (x) = −N u, ∇HN (x) = 0}. Gaussian regression gives E[HN (σ) | Ex,u ] = −N tp u,

(4.22)

Var(HN (σ) | Ex,u ) = N vp (t).

(4.23)

In geodesic normal coordinates whose first tangent direction points toward σ, one also has the rank-at-most-one covariance Cov(∇2 HN (x), HN (σ) | Ex,u ) = ap (t)e1 eT 1. These identities are verified in Appendix A.

(4.24)

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

Lemma 4.4 (Exact marked Kac–Rice identity). For Borel sets A ⊂ R and I ⊂ (−1, 1), KN (A, I) = κN,p

−N u2 /2

(

N β2 νN (t) exp N βt u + vp (t) 2 I

)

× E det KN (u) − βap (t)e1 eT 1

dt du.

(4.25)

Equivalently, with E| det(KN (u) − βap (t)e1 eT 1 )| , E| det KN (u)|

DN (u, t) =

(4.26)

one has KN (A, I) =

νN (t)eN βt u+N β vp (t)/2 DN (u, t) dt du.

ρN (u)

(4.27)

Proof. If (X, Y ) is jointly Gaussian and F ≥ 0, Gaussian exponential tilting gives 2

= eN βt u+N β vp (t)/2 E| det(KN (u) − βap (t)e1 eT 1 )|.

(4.28)

The marked one-point Kac–Rice formula, isotropy, and (2.3) now yield (4.25). Substituting (4.13) gives (4.27). □ Corollary 4.5 (Uniform rank-one stability). For every L > 0 and compact J ⋐ (−1, 1), 1 lim sup sup log DN (u, t) ≤ 0. (4.29) N N →∞ |u|≤L t∈J

Proof. Cofactor expansion gives, for every scalar b, (1) | det(K − be1 eT |. 1 )| ≤ | det K| + |b| | det K

Since supt∈J |βap (t)| < ∞, the assertion follows from (4.17) and (4.18).

□

Lemma 4.6 (Uniform energy-tail bound). For every compact J ⋐ (−1, 1), there is C < ∞ such that, for all N ≥ 3, u ∈ R, and t ∈ J, N −1 E| det(KN (u) − βap (t)e1 eT . 1 )| ≤ [C(1 + |u|)]

(4.30)

1 log KN ({u : |u| > L}, J) = −∞. N

(4.31)

Moreover, lim lim sup

L→∞ N →∞

TAEGYUN KIM

Proof. For an n × n matrix A, the arithmetic–geometric mean inequality for its singular values gives ∥A∥F n . | det A| ≤ √ n Under (4.1), √ d 2 ∥Gn ∥2F = χ2n(n+1)/2 , (E∥Gn ∥nF )1/n ≤ C0 n. n Minkowski’s inequality in Ln , (4.2), and the boundedness of ap on J prove (4.30). R Also, 0 ≤ vp (t) ≤ 1, |tp u| ≤ |u|, J νN (t) dt ≤ 1, and κN,p ≤ eC1 N . Hence (4.25) implies C2 N

KN ({|u| > L}, J) ≤ e

(

u2 exp N − + β|u| + log(1 + |u|) 2 |u|>L

du.

For z ≥ 0, z2 + 2β 2 , 8 Consequently, for L > 0, βz ≤

log(1 + z) ≤ z ≤ (

z2 + 2. 8

L2 4 exp N C3 − KN ({|u| > L}, J) ≤ NL 4

(4.32) □

which proves (4.31).

Proposition 4.7 (Marked Kac–Rice bound). Fix p ∈ {3, 4, . . .}, β > 0, and a compact interval J ⋐ (−1, 1). For every fixed closed interval I ⊂ J of positive length, 1 (4.33) lim sup log KN (R, I) ≤ sup Φp,β (u, t). N N →∞ u∈R, t∈I Since J is arbitrary, it follows in particular that, for every q ∈ (0, 1), 1 lim lim sup log EMN (q, η) ≤ sup Φp,β (u, q). (4.34) η↓0 N →∞ N u∈R Proof. Fix L > 0. Uniformly for |u| ≤ L and t ∈ J, 1 log ρN (u) = Σp (u) + o(1), N 1 1 log νN (t) = log(1 − t2 ) + o(1), N 2 1 log DN (u, t) ≤ o(1), N by Lemma 4.3 and Corollary 4.5, Stirling’s formula, and (2.3). Applying the elementary Laplace upper bound to (4.27) gives 1 lim sup log KN ([−L, L], I) ≤ sup Φp,β (u, t). N →∞ N |u|≤L t∈I

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

The energy-tail estimate (4.31) then removes the truncation and proves (4.33). Finally, Ω(x) = O(log(1 + |x|)) as |x| → ∞. Thus, locally uniformly for t ∈ (−1, 1), the function Φp,β (u, t) tends to −∞ quadratically as |u| → ∞. It follows by a common compact truncation that t 7→ supu Φp,β (u, t) is continuous. Taking I = [q − η, q + η] ⊂ (−1, 1) and then η ↓ 0 proves (4.34). □ Remark 4.8. For every collection A of critical points, !

ZN,β

[

B(x, q, η)

≤

x∈A

ZN,β (B(x, q, η)) ≤ MN (q, η).

(4.35)

x∈A

No separation assumption is needed for this upper bound. 5. A general-p marked transition The marked variational problem has a sharp transition throughout the high-temperature side. This calculation both implies the analytic part of Theorem 1.1 and locates the obstruction for p ≥ 4. For p ≥ 3, set p−2 , s∗ = p−1

u∞,p = 2

p−1 , p

βsh (p) =

(p − 1)p−1 . p(p − 2)p−2

(5.1)

For 0 < β ≤ βsh (p), define √ β2 ∆p,β (s) = sup Φp,β (u, s) − , 2 u∈R

0 ≤ s < 1.

(5.2)

Proposition 5.1 (General-p marked sign law). Fix p ≥ 3 and 0 < β ≤ βsh (p). Then   0 ≤ s < s∗ ,  1, sgn ∆p,β (s) = 0, (5.3) s = s∗ ,   −1, s < s < 1. ∗ At s = s∗ , the maximizing energy is unique and equals 2(p − 1) p/2 u∗p,β = βs∗ . p−2 In particular, u∗p,βsh (p) = u∞,p .

(5.4)

Proof. Direct integration in (4.3) gives  2  x − 1,  4 2

|x| ≤ 2,

! √ √ Ω(x) = x2 1 |x| x2 − 4 |x| + x2 − 4    − − + log ,  4 2 4 2

Consequently, on the quadratic branch, 1 p−2 2 Σp (u) = log(p − 1) − u , 2 4(p − 1)

(5.5) |x| ≥ 2.

|u| ≤ u∞,p .

(5.6)

TAEGYUN KIM

For u = u∞,p cosh a, a ≥ 0, the same formula gives −Σ′p (u) =

(p − 2) cosh a + p sinh a p . p(p − 1)

Together with (5.6), this shows that −Σ′p is strictly increasing on [0, ∞), with matching one-sided values at u∞,p . By evenness, Σp is strictly concave; it is also coercive by (5.5). Hence, after adding any √ linear mark, the maximizer is unique. Since the coefficient of u in Φp,β (u, s) is nonnegative, the unique maximizer is nonnegative. Whenever it lies in the quadratic branch, it is up,β (s) =

2(p − 1) p/2 βs , p−2

(5.7)

and substitution in (4.6) gives pβ 2 p−1 p − 1 1 log (p − 1)(1 − s) + s s−1 . 2 2 p−2

∆p,β (s) =

(5.8)

The identity up,β (s) β s p/2 = (5.9) u∞,p βsh (p) s∗ shows that (5.8) applies throughout 0 ≤ s ≤ s∗ . It immediately gives ∆p,β (s∗ ) = 0 and (5.4). We next prove positivity for s < s∗ . Put s n = p − 2, x= . s∗

At β = βsh (p), equation (5.8) becomes 2∆p,βsh (p) (s) = log 1 + n(1 − x) − nxn+1 (1 − x).

(5.10)

For t = 1 − x ∈ (0, 1), log(1 + nt) >

nt > nt(1 − t)n+1 . 1 + nt

The second inequality follows from (1−t)−(n+1) > 1+nt. Thus the right-hand side of (5.10) is positive; the case t = 1 is immediate from log(1 + n) > 0. For s < s∗ , the coefficient of β 2 in (5.8) is negative, and hence ∆p,β (s) ≥ ∆p,βsh (p) (s) > 0. It remains to treat s > s∗ . We first work at β = βsh (p). By (5.9), the formal quadratic stationary point then lies strictly beyond u∞,p . Strict concavity therefore places the actual maximizer in the outer branch. Write s = s∗ x and parametrize it by u = u∞,p cosh a,

y = ea .

Formula (5.5) yields Σp (u∞,p cosh a) =

1 p−2 p − 1 2a 1 log(p − 1) − +a− e + e−2a . 2 2p 2p 2p

(5.11)

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

The critical-point equation for the maximizing a becomes (n + 1)y − y −1 = nx(n+2)/2 .

(5.12)

Set L = n + 1 − nx,

z = xn/2 .

Since 1 < x < (n + 1)/n, we have L > 0. Envelope differentiation, followed by substitution of (5.12), gives d n ∆p,βsh (p) (s∗ x) = (y − zL)(zL − y −1 ). dx 2L

(5.13)

For x > 1, L (zL) = nz − 1 < 0, 2x so zL decreases from one. Since y > 1, this gives y − zL > 0. To determine the other factor, define

′

f (v) = (n + 1)v − v −1 − nx(n+2)/2 . The function f is strictly increasing, f (y) = 0, and a direct calculation gives zL f (zL)−1 = (n + 1)(1 − xn L) > 0,

because (xn L)′ = n(n + 1)xn−1 (1 − x) < 0, so xn L decreases from one on this interval. Hence y < (zL)−1 , so zL−y −1 < 0. It follows from (5.13) that ∆p,βsh (p) (s) < 0,

s > s∗ .

Finally fix s > s∗ and let up,β (s) be the maximizing energy. The envelope theorem gives h

∂β ∆p,β (s) = sp/2 up,β (s) − βs(p−2)/2 p − (p − 1)s

(5.14)

This derivative is strictly positive. In the quadratic branch, the claim reduces by (5.7) to s > s∗ . In the outer branch, up,β (s) ≥ u∞,p , while s 7−→ s(p−2)/2 p − (p − 1)s

is strictly decreasing for s > s∗ , and (p−2)/2

2βsh (p)s∗

= u∞,p .

Since β ≤ βsh (p), the bracket in (5.14) is again positive. Therefore ∆p,β (s) ≤ ∆p,βsh (p) (s) < 0, which completes the proof.

□

By Lemma 2.3, the reference value β 2 /2 in (5.2) is the full limiting free energy in this range.

TAEGYUN KIM

Proof of Theorem 1.1. For q ≤ 2−1/2 , Proposition 3.3 rules out positive √ complexity. For q > s∗ , Proposition 5.1 gives a fixed gap γ=

β2 − sup Φp,β (u, q) > 0. 2 u∈R

Continuity and coercivity give a fixed η0 > 0 such that sup Φp,β (u, t) ≤ u∈R |t−q|≤η0

β 2 3γ − . 2 4

After decreasing η0 if necessary so that [q −η0 , q +η0 ] ⋐ (0, 1), Proposition 4.7 implies that, for all sufficiently large N , (

EMN (q, η0 ) ≤ exp N

β2 γ − 2 2

Markov’s inequality and (4.35) therefore show, with probability tending to one, that the union of any critical-point bands of width at most η0 has free energy at most β 2 /2 − γ/4. Since every admissible shattering width ηN eventually satisfies ηN ≤ η0 , while Lemma 2.3 gives FN (β) → β 2 /2, this contradicts condition (4) in Definition 2.1. p Finally, when p = 3, one has (p − 2)/(p − 1) = 2−1/2 , so the two ranges in (1.3) exhaust q ∈ (0, 1). Also βsh (3) = βsh , equivalently Tsh (3) = Tsh , which proves the stated specialization. The additional criterion (1.4) and its all-overlap consequence (1.5) are exactly Proposition 3.6. □ Remark 5.2 (The geometric window for p ≥ 4). The only overlap range not excluded by the small-overlap and marked Kac–Rice arguments is p−2 1 <s≤ , 2 p−1

equivalently

1 √ <q≤ 2

p−2 . p−1

(5.15)

For p = 3, this window is empty. For p ≥ 4, it is nonempty, and at the contact point p−3 2s∗ − 1 = > 0. (5.16) p−1 Thus the thermodynamic contact occurs where the universal obstruction based on negative overlaps no longer applies. The positive margin does not prove shattering: it only shows that the N + 1 cardinality bound supplies no control there. The spherical-code argument in Proposition 3.6 resolves every point in this window for which β 2 < −2RLP (2q 2 − 1) − log(1 − q 2 ). (5.17) √ In particular, it resolves the whole window for β ≤ log 2. Resolving the remaining part requires more than one-point marked Kac–Rice, universal latitude geometry, and the entropy estimate used here.

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

If r = 0, condition (2) in Definition 2.1 itself forces strictly negative pairwise center overlaps. Then Lemma 3.2 gives at most N + 1 centers for every p, every temperature, and every q. Thus the unresolved window concerns positive separation parameters r. 6. Endpoint optimization for the pure 3-spin model Although Proposition 5.1 already implies the endpoint inequality, an explicit p = 3 optimization identifies √ the order of the tangency. We now specialize to p = 3 and β = βsh = 2/ 3. Set r

u∞ = 2

2 = 3

8 . 3

(6.1)

Specializing (4.4) and using (5.5), 1 1 u2 Σ3 (u) = + log 2 − +Ω 2 2 2

3 u . 2

(6.2)

In particular, for 0 ≤ u ≤ u∞ , Σ3 (u) =

1 u2 log 2 − . 2 8

(6.3)

Write s = q 2 . From (4.6), √ 2 2 1 Ψ(u, s) := Φ3,βsh (u, s) = Σ3 (u) + √ s3/2 u + log(1 − s) + (1 − 3s2 + 2s3 ). 2 3 3 (6.4) Proposition 6.1 (Endpoint variational inequality). For every s ∈ [1/2, 1), 2 sup Ψ(u, s) ≤ . 3 u∈R

(6.5)

Equality in (6.5) holds if and only if s = 1/2; at this value, the unique maximizing energy is u = u∞ . In particular, for every q ∈ (2−1/2 , 1), 2 sup Φ3,βsh (u, q) < . 3 u∈R

(6.6)

Proof. Since Σ3 is even and the coefficient of u in (6.4) is positive, it suffices to maximize over u ≥ 0. For 0 ≤ u ≤ u∞ , (6.3) gives ∂u Ψ(u, s) = −

u 2 + √ s3/2 . 4 3

(6.7)

At u = u∞ , this derivative is nonnegative for s ≥ 1/2, with equality only at s = 1/2. Hence no maximizer lies in [0, u∞ ). For u ≥ u∞ , write u = u∞ cosh a,

a ≥ 0.

(6.8)

Since

TAEGYUN KIM

3/2 u = 2 cosh a, (5.5) yields 1 1 1 1 Σ3 (u∞ cosh a) = log 2 − + a − e2a + e−2a . 2 6 3 6

(6.9)

Define

2 D(a, s) = Ψ(u∞ cosh a, s) − . 3 Substituting (6.9) into (6.4), we obtain 1 1 1 1 D(a, s) = log 2(1 − s) − + a − e2a + e−2a 2 √ 6 3 6 4 4 2 3/2 s cosh a − 2s2 + s3 . + 3 3 a Let y = e . Differentiation gives the exact factorization √ (y − 1)(y + 1) 2y 2 − 2 2s3/2 y − 1 ∂a D(a, s) = − . 3y 2

(6.10)

(6.11)

For s > 1/2, D(·, s) first increases and then decreases on [0, ∞). Its unique maximizer as > 0 is determined by √ 2e2as − 2 2s3/2 eas − 1 = 0. (6.12) At s = 1/2, the maximizing point is a1/2 = 0. Equivalently, √ cosh as + 3 sinh as = 2 2s3/2 .

(6.13)

Let G(s) = D(as , s), A direct substitution into (6.10) gives

s ∈ [1/2, 1).

G(1/2) = 0. For s ∈ (1/2, 1), the envelope theorem and (6.10) give √ 1 . G′ (s) = 2 2 s1/2 cosh as + 4s2 − 4s − 2(1 − s) Solving (6.12) for eas and then for cosh as gives √ 3 1 + s3 − s3/2 √ cosh as = . 2 2 Therefore p 1 G′ (s) = 3 s + s4 + 3s2 − 4s − . 2(1 − s) Set 1 R(s) = 4s − 3s2 + . 2(1 − s) √ Then G′ (s) = 3 s + s4 − R(s). An exact algebraic identity is R(s)2 − 9(s + s4 ) = −

(2s − 1)3 (12s2 − 14s + 1) . 4(1 − s)2

(6.14) (6.15)

(6.16)

(6.17)

(6.18)

(6.19)

(6.20)

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

For s ∈ (1/2, 1), 12s2 − 14s + 1 < 0, √ because its roots are (7 ± 37)/12, with the upper root larger than one. Thus the right-hand side of (6.20) is strictly positive. Since R(s) > 0, p

R(s) > 3 s + s4 , and hence G′ (s) < 0,

s ∈ (1/2, 1).

(6.21)

Together with (6.15), this proves G(s) < 0 for every s > 1/2. Equality occurs only at s = 1/2, where a = 0, equivalently u = u∞ . □ Remark 6.2 (Tangency at the geometric threshold). The equality point in Proposition 6.1 is 1 q=√ , u = u∞ . 2 This is exactly the boundary at which Lemma 3.1 changes from requiring negative pairwise center overlaps to permitting nonnegative ones. Thus the thermodynamic exponent reaches the full free energy only at the point where exponential disjointness is geometrically impossible. Writing 2 γ(q) = − sup Φ3,βsh (u, q), 3 u∈R the contact is fourth order. Indeed, Taylor expansion of (6.18) at s = 1/2 gives, as q ↓ 2−1/2 , 5 !

1 q − 2

16 1 = q−√ 3 2

1 5 +O q−√ . +O 2 (6.22) This slow closing of the gap is another reason that the fixed-q quantifier in Definition 2.1 matters. 4 2 1 γ(q) = q − 3 2

7. Discussion and scope For every 0 < β ≤ βsh , the marked exponent touches the full free energy at q 2 = 1/2 and is strictly below it for every q 2 > 1/2. The deterministic obstruction rules out positive complexity at and below the contact latitude. At β = βsh , the optimizing energy reaches u∞ and the contact becomes fourth order, as shown in Remark 6.2. Thus the pure 3-spin landscape is not shattered throughout T ≥ Tsh . The joint proof of Theorems 2.5 and 8.4 in [4, Section 5] assumes the strict inequality β > βsh . In the p = 3 discussion in [4, Section 8], this yields 1 q∗∗ (β) > √ , 2

TAEGYUN KIM

which provides the positive geometric margin used in the disjointness argument. At equality, 1 q∗∗ (βsh ) = √ , 2 and the deterministic argument in Section 3 shows that an exponentially large disjoint family is impossible. The marked Kac–Rice calculation shows, in addition, that moving to any fixed larger q loses a strictly positive amount of free energy. Thus the strict inequality in the proof reflects a real endpoint transition. The coincidence between the marked transition and the geometric cutoff is special to p = 3. For general p, write qcont for the contact latitude. Then 2 qcont = s∗ =

p−2 , p−1

so for p ≥ 4, 2 2qcont −1=

p−3 > 0. p−1

Hence the present argument leaves the range 1/2 < q 2 ≤ s∗ unresolved only where the reverse weak inequality to (5.17) √ √ holds. This remaining set is contained in log 2 < β ≤ βsh (p); for β ≤ log 2, Proposition 3.6 supplies an all-overlap obstruction. To our knowledge, the literal dynamical endpoint for p ≥ 4 remains open under Definition 2.1; Theorem 1.1 records the ranges that the present method does settle. This paper settles fixed-overlap, critical-point-band non-shattering at and above the dynamical temperature for the pure 3-spin model. It does not settle the strict interior Ts < T < Tsh , the lower endpoint T = Ts , or the lower-temperature regime. Accordingly, any p = 3 shattered regime under Definition 2.1 must lie strictly below Tsh . Recent works study related but different notions. For the Ising pure p-spin model at large p, Gibbs-measure shattering was proved in [11] and extended down to the conjectural dynamical scale, to leading order as p → ∞, in [8]. Those results use configuration-space clusters and overlap-gap methods rather than the fixed-overlap critical-point bands of Definition 2.1. El Alaoui, Montanari, and Sellke [9] prove a Gibbs-measure shattering decomposition for pure spherical models when p is sufficiently large. Their clusters need not be bands around critical points, so that result neither covers p = 3 nor decides the literal endpoint question under Definition 2.1. The Franz–Parisi potential was introduced in [10]. Auffinger, El Alaoui, and Sellke [2, Section 5] discuss the strict spherical criterion s β 2 ξ ′ (s) > , s ∈ (0, 1). 1−s For the pure model ξ(s) = sp , this becomes β 2 psp−2 (1 − s) > 1.

NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

For every 0 < β ≤ βsh (p), the reverse weak inequality holds for all s ∈ (0, 1); equality occurs only when β = βsh (p) and s = (p − 2)/(p − 1). For p = 3, this is s = 1/2 = q 2 , matching the contact point in Proposition 5.1. This remains only a consistency check: equivalence between that Franz–Parisi-potential criterion and Definition 2.1 is not presently established. Acknowledgments The author is deeply grateful to Gérard Ben Arous for many helpful discussions and for his generous guidance on spherical spin glasses, shattering, and many related topics. This work was supported by Jang Young Sil Fellow Program at KAIST. Appendix A. Gaussian regression at a critical point For completeness, we verify Lemma 4.1 and (4.22)–(4.24). By rotational invariance, take p √ √ x = N eN , σ = N teN + 1 − t2 e1 . View HN as the restriction to SN of the homogeneous Gaussian polynomial on RN with covariance EHN (z)HN (w) = N 1−p (z · w)p . amb H (x). Homogeneity For tangent indices 1 ≤ i, j ≤ N − 1, let Bij = ∂ij N √ N −1 and the second fundamental form of S ( N ) give p (∇2 HN (x))ij = Bij − HN (x)δij . (A.1) N Differentiating the ambient covariance shows

p(p − 1) (δik δjℓ + δiℓ δjk ), N Cov(Bij , HN (x)) = Cov(Bij , ∂k HN (x)) = 0. Cov(Bij , Bkℓ ) =

(A.2) (A.3)

Hence B is independent of (HN (x), ∇HN (x)), and under HN (x) = −N u, (A.1) is a GOE matrix with mean puIN −1 and covariance (4.10). With the normalization (4.1), this is exactly KN (u) from (4.2), proving Lemma 4.1. Let ∂i denote differentiation at x in geodesic normal coordinates in the tangent direction ei , 1 ≤ i ≤ N − 1. Differentiating C(z, w) = EHN (z)HN (w) = N R(z, w)p gives Var(HN (x)) = N,

(A.4) p

Cov(HN (x), HN (σ)) = N t ,

(A.5)

Cov(∂i HN (x), ∂j HN (x)) = pδij ,

(A.6)

Cov(HN (σ), ∂i HN (x)) = ptp−1 σi .

(A.7)

TAEGYUN KIM

Only the i = 1 term in (A.7) is nonzero. Since HN (x) is independent of ∇HN (x), Gaussian regression immediately yields E[HN (σ) | HN (x) = −N u, ∇HN (x) = 0] = −N tp u, and Var(HN (σ) | HN (x), ∇HN (x)) = N − N t2p −

−1 1 NX Cov(HN (σ), ∂i HN (x))2 p i=1

= N 1 − t2p − pt2p−2 (1 − t2 ) ,

which is (4.23). For the Riemannian Hessian, a second differentiation gives σi σj 2 − ptp δij . Cov(∂ij HN (x), HN (σ)) = p(p − 1)tp−2 N The curvature term is the second term on the right. Similarly, 2 Cov(∂ij HN (x), HN (x)) = −pδij .

(A.8)

(A.9)

The Hessian is independent of the gradient. Conditioning on HN (x) therefore cancels the identity term in (A.8): 2 Cov(∂ij HN (x), HN (σ) | HN (x), ∇HN (x)) σi σj = p(p − 1)tp−2 . N Since σ12 /N = 1 − t2 and σi = 0 for 2 ≤ i ≤ N − 1, this is precisely (4.24).

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NON-SHATTERING AT AND ABOVE THE DYNAMICAL TEMPERATURE

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Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

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