on second moments and extremal critical points of p-spin glasses

Overview

Auffinger, Ben Arous, and Cerny computed the first moment of the number of critical points in the pure spherical $p$-spin model using Kac-Rice and GOE asymptotics (Auffinger, Ben Arous & Černý, 2013). Subag’s second-moment paper answers the next probabilistic question: in the low-energy band, is this first moment typical or only annealed? For the total number of critical points below level $Nu$, the answer is typical throughout

$$u\in (-E_0(p),-E_{\infty }(p))$$

(Subag, 2017).

Subag and Zeitouni then study the still finer $O(1)$ window around the ground state. After the right logarithmic centering, the near-minimal critical values converge to a Poisson point process with exponential intensity, and the centered ground-state energy converges to a negative Gumbel law (Subag & Zeitouni, 2017). In this sense, the extreme bottom of the pure spherical $p$-spin landscape behaves like the Random Energy Model.

🏷️ Setup and Notation

We use the pure spherical $p$-spin Hamiltonian

$$H_N(\sigma )=\frac{1}{N^{(p-1)/2}}\sum _{i_1,\dots ,i_p=1}^N{J}_{i_1,\dots ,i_p}{\sigma }_{i_1}\cdots {\sigma }_{i_p},\sigma \in S^{N-1}(\sqrt{N}),$$

with $p\ge 3$ fixed and independent standard Gaussian coefficients $J_{i_1,\dots ,i_p}$. The covariance is

$$\mathbb{E}{H}_N(\sigma )H_N({\sigma }^{\prime })=NR(\sigma ,{\sigma }^{\prime }{)}^p,R(\sigma ,{\sigma }^{\prime })=\frac{1}{N}⟨\sigma ,{\sigma }^{\prime }⟩.$$

All gradients and Hessians below are intrinsic to the sphere. For a Borel set $B\subset \mathbb{R}$, write

$${\mathcal{C}}_N(B)=\{\sigma :\nabla H_N(\sigma )=0,H_N(\sigma )/N\in B\},{\mathrm{Crt}}_N(B)=|{\mathcal{C}}_N(B)|.$$

The first-moment theorem of Auffinger, Ben Arous, and Cerny says

$$\underset{N\to \infty }{\lim }\frac{1}{N}\log \mathbb{E}{\mathrm{Crt}}_N((-\infty ,u))={\Theta }_p(u),$$

where the zero of ${\Theta }_p$ occurs at $-E_0(p)$, and the spectral-edge threshold is

$$E_{\infty }(p)=2\sqrt{\frac{p-1}{p}}.$$

The first moment immediately gives absence below $-E_0(p)$: if $u<-E_0(p)$, then the expected number of critical points below $Nu$ decays exponentially, so Markov’s inequality rules them out with high probability. The hard direction is abundance: if $u>-E_0(p)$ and the mean is exponentially large, is a typical landscape also populated there?

🏷️ Second Moment: What Is Proved

Subag proves that for every

$$u\in (-E_0(p),-E_{\infty }(p)),$$

the total low-energy critical-point count satisfies

$$\frac{\mathbb{E}\left[{\mathrm{Crt}}_N((-\infty ,u){)}^2\right]}{{\left(\mathbb{E}{\mathrm{Crt}}_N((-\infty ,u))\right)}^2}\to 1.$$

Therefore

$$\frac{{\mathrm{Crt}}_N((-\infty ,u))}{\mathbb{E}{\mathrm{Crt}}_N((-\infty ,u))}\to 1$$

in $L^2$ and hence in probability.

Interpretation

In the band $(-E_0(p),-E_{\infty }(p))$, the annealed total complexity is the typical total complexity. The first-moment exponent from on random matrices and complexity of spin glasses is not a rare-event artifact there.

Subag also proves a weaker but still useful statement for every $u>-E_0(p)$: the second moment matches the square of the first moment on the exponential scale,

$$\underset{N\to \infty }{\lim }\frac{1}{N}\log \mathbb{E}\left[{\mathrm{Crt}}_N((-\infty ,u){)}^2\right]=2{\Theta }_p(u).$$

This does not by itself give $L^2$ concentration, but it rules out an exponentially large second-moment gap.

Total count, not fixed index

The theorem above concerns ${\mathrm{Crt}}_N$, the total count over all indices. The paper explains how one would add fixed-index restrictions to the two-point Kac-Rice formula, but does not complete that analysis.

🏷️ Pair Kac-Rice and the Overlap Variable

The second moment counts ordered pairs of critical points. The right geometric coordinate for pairs is the overlap

$$r=R(\sigma ,{\sigma }^{\prime })=\frac{1}{N}⟨\sigma ,{\sigma }^{\prime }⟩.$$

For an overlap set $I_R\subset [-1,1]$, define the pair count

$$[{\mathrm{Crt}}_N(B,I_R){]}_2=\#\{(\sigma ,{\sigma }^{\prime }):\sigma ,{\sigma }^{\prime }\in {\mathcal{C}}_N(B),R(\sigma ,{\sigma }^{\prime })\in I_R\}.$$

Then

$$\mathbb{E}[{\mathrm{Crt}}_N(B,[-1,1]){]}_2=\mathbb{E}{\mathrm{Crt}}_N(B{)}^2.$$

This is an ordered-pair count; when $I_R=[-1,1]$, diagonal pairs are included, so the displayed identity is literal.

Two-point Kac-Rice expresses this expectation as an integral over $r$. After conditioning on the values and gradients at the two points, the two Hessians become correlated shifted GOE matrices plus low-rank perturbations. Schematically,

$${\nabla }^2{H}_N({\sigma }_i)\approx X_i(r)-\text{shift}(u_i)I+\text{low-rank correction},i=1,2,$$

where $X_1(r)$ and $X_2(r)$ are correlated GOE matrices.

The first-moment computation only had one shifted GOE determinant. The second moment has a product of two such determinants, and the matrices are correlated when $r\ne 0$. This is the main new difficulty.

🏷️ Exponential Scale: Why Orthogonality Wins

The two-point Kac-Rice formula leads to a variational upper bound

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\log \mathbb{E}[{\mathrm{Crt}}_N(B,I_R){]}_2\le \underset{r\in I_R}{\sup }\underset{u_1,u_2\in B}{\sup }{\Psi }_p(r,u_1,u_2).$$

The function ${\Psi }_p$ has three conceptual contributions.

• The volume of pairs with overlap $r$ gives an entropy term. Most pairs on a high-dimensional sphere are nearly orthogonal, so this term favors $r=0$.
• The joint Gaussian density of the two energy values gives a quadratic cost depending on $u_1,u_2,r$.
• The logarithms of the two Hessian determinants become linear statistics of GOE eigenvalues. Their exponential-scale behavior is controlled by the semicircle law and the large deviation principle for empirical eigenvalue measures.

The decisive calculus result is that, in the low-energy range relevant for the second-moment theorem, the supremum is attained at

$$r=0,u_1=u_2=u.$$

This says that the dominant contribution to the second moment comes from pairs of critical points with asymptotically zero overlap. Correlated pairs do not create a second-moment explosion.

Proof spine: exponential matching

The determinant term is handled by writing

$$\frac{1}{N}\log |\det (X-aI)|=\int \log |x-a|d{L}_N(x),$$

where $L_N$ is the empirical spectral measure. Since $L_N$ satisfies a large deviation principle at speed $N^2$, while the complexity problem lives at speed $N$, the main determinant contribution is the semicircle linear statistic at the exponential scale. The paper still has to control truncations and edge-tail events to make the bound uniform. The remaining optimization is over the overlap $r$ and the energy values $u_1,u_2$.

🏷️ From Exponential Matching to $L^2$ Matching

Exponential matching is not enough for

$$\frac{\mathbb{E}{\mathrm{Crt}}_N((-\infty ,u){)}^2}{(\mathbb{E}{\mathrm{Crt}}_N((-\infty ,u)){)}^2}\to 1.$$

It only says the logarithms agree up to $o(N)$.

The final $O(1)$ refinement uses two additional facts.

First, energies away from the upper endpoint of the interval are negligible. Since ${\Theta }_p$ is strictly increasing below $0$, the count below $u$ is exponentially dominated by a thin window near $u$.

Second, overlaps outside a small neighborhood of $0$ are negligible. The variational analysis gives a strict gap away from $r=0$, so only nearly orthogonal pairs remain.

Thus the proof reduces to a very local estimate with

$$u_1,u_2\in (u-{\epsilon }_N,u),r\in (-{\rho }_N,{\rho }_N),$$

where ${\epsilon }_N,{\rho }_N\to 0$. In this regime the correlated GOE determinant expectation is asymptotically the product of two independent determinant expectations. This proves the full second-moment ratio.

🏷️ Consequence for the Ground State

The first moment already proves that, with high probability, no critical point lies below

$$-N(E_0(p)+\epsilon ).$$

Subag’s concentration result proves the complementary abundance statement: for every $\epsilon >0$, with high probability there are critical points below

$$-N(E_0(p)-\epsilon ).$$

Therefore the normalized ground-state energy satisfies

$$\frac{1}{N}\underset{\sigma \in S^{N-1}(\sqrt{N})}{\min }{H}_N(\sigma )\to -E_0(p)$$

almost surely, hence also in probability. This gives a route to the leading ground-state value using critical-point moments, rather than relying on the Parisi formula.

🏷️ Extremal Process: The $O(1)$ Scale

Subag and Zeitouni refine the ground-state statement from the $N$ scale to the $O(1)$ scale. Define

$$c_p={\Theta }_p^{\prime }(-E_0(p))$$

and use the centering

$$m_N=-E_0(p)N+\frac{\log N}{2c_p}-K_0,$$

where $K_0$ is an explicit constant.

Let $C_N$ be the set of critical points and set

$${\xi }_N=(1+{\iota }_p{)}^{-1}\sum _{\sigma \in C_N}{\delta }_{H_N(\sigma )-m_N},{\iota }_p=\frac{1+(-1{)}^p}{2}.$$

The factor $(1+{\iota }_p{)}^{-1}$ removes the double counting caused by the antipodal symmetry when $p$ is even.

The main theorem is

$${\xi }_N\Rightarrow \mathrm{PPP}(e^{c_{p}x}dx),$$

where convergence is in the vague topology on point processes.

Why this implies a Gumbel law

If $\xi$ is a Poisson point process on $\mathbb{R}$ with intensity $e^{c_{p}x}dx$, then

$$\mathbb{P}\{\xi ((-\infty ,x))=0\}=\exp \left(-\frac{1}{c_p}{e}^{c_{p}x}\right).$$

Therefore

$$\underset{\sigma }{\min }{H}_N(\sigma )-m_N$$

converges to a negative Gumbel-type random variable.

Near this extremal scale, all relevant critical points are local minima with high probability. Thus the extremal process can be read as the point process of the deepest minima.

🏷️ Why the Poisson Limit Appears

The proof has two complementary inputs.

The first input is moment control. A sharp first-moment estimate shows that the intensity measure of ${\xi }_N$ is asymptotically

$$e^{c_{p}x}dx.$$

The second-moment machinery from Subag controls overlaps between near-extremal critical points, showing that distinct near-ground-state critical points have overlaps tending to $0$.

The second input is a perturbation-invariance argument. Let $H_N^{\prime }$ be an independent copy of the Hamiltonian and consider the slightly perturbed field

$$H_N^{+}(\sigma )=H_N(\sigma )+N^{-1/2}{H}_N^{\prime }(\sigma ).$$

As a Gaussian field,

$$H_N^{+}\stackrel{d}{=}\sqrt{\frac{N+1}{N}}{H}_N.$$

So the extremal process of $H_N^{+}$ has the same limiting law as that of $H_N$, shifted by a deterministic amount.

On the other hand, if the near-extremal critical points are asymptotically orthogonal, then the perturbation values $H_N^{\prime }(\sigma )$ at those points behave asymptotically like independent Gaussians. A local quadratic approximation shows that each near-minimum critical point of $H_N$ moves to a nearby critical point of $H_N^{+}$, and its critical value is shifted by approximately

$$W-C_0,W\sim N(0,1).$$

Thus any subsequential limit of ${\xi }_N$ must be invariant under independently shifting its atoms by Gaussian increments, up to a deterministic recentering. Liggett’s characterization and the Choquet-Deny theorem first force such a limit to be a Cox process whose intensity density is almost surely either constant or exponential of slope $c_p$. The first-moment asymptotics and tightness at the lower tail then identify the non-random intensity as exactly

$$e^{c_{p}x}dx.$$

Proof spine: from moments to Poisson

The moment estimates give tightness and rule out clusters at nonzero overlap. The perturbation argument identifies the only possible subsequential limits. The first moment fixes the intensity. Together these three pieces upgrade low-energy concentration into a full extremal-process theorem.

🏷️ What These Papers Add

The two papers answer different levels of the same question.

Scale	Result	Method
Exponential count below $Nu$	${\mathrm{Crt}}_N((-\infty ,u))$ concentrates around its mean for $u\in (-E_0,-E_{\infty })$	two-point Kac-Rice, overlap variational problem, GOE determinant asymptotics
Ground-state leading order	$N^{-1}\min H_N\to -E_0$	first moment for absence plus second moment for abundance
Ground-state $O(1)$ window	critical values near $m_N$ converge to $\mathrm{PPP}(e^{c_{p}x}dx)$	sharp first moment, second-moment overlap control, perturbation invariance
Minimum fluctuation	$\min H_N-m_N$ has negative Gumbel limit	empty interval probability for the Poisson process

Conceptually, the first paper says that the annealed complexity is reliable in the low-energy pure model. The second says that at the very bottom the deepest minima behave almost like independent extreme samples.

🔗 See Also

• on random matrices and complexity of spin glasses --- The first-moment Kac-Rice/GOE computation is the input for both Subag’s second-moment theorem and the Subag-Zeitouni extremal-process theorem.
• on Slepian’s lemma and Gaussian comparison --- The perturbation argument uses Gaussian structure and approximate independence at vanishing overlap.
• on Dudley’s Theorem --- Useful background for thinking about Gaussian fields on high-dimensional index sets, though the present results count critical points rather than suprema.
• on Ising models --- Provides the statistical-mechanics contrast between discrete spin systems and the continuous spherical model.

📚 References

🐻  Auffinger, A., Ben Arous, G. & Černý, J. 2013. Random Matrices and Complexity of Spin Glasses. Communications on Pure and Applied Mathematics 66(2), 165–201.

🐻  Subag, E. 2017. The complexity of spherical p-spin models—a second moment approach. The Annals of Probability 45(5), 3385–3450.

🐻  Subag, E. & Zeitouni, O. 2017. The extremal process of critical points of the pure p-spin spherical spin glass model. Probability Theory and Related Fields 168(3–4), 773–820.