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The extremal process of critical points of the pure p-spin spherical spin glass model


Recently, sharp results concerning the critical points of the Hamiltonian of the p-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points—that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.

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  1. 1.

    A decorated Poisson point process is a process whose law is obtained from a Poisson point process by replacing each of the atoms by an independent copy of some point process (called the “decoration”). An SDPPP is simply a decorated Poisson point process, shifted by an independent random variable.

  2. 2.

    For fixed open \(J_N=J\), by the Portmanteau theorem \(\liminf _{N\rightarrow \infty } \mathbb {E}\{ \left( \xi _{N}\left( J\right) \right) ^{2}\}\ge \mathbb {E}\{ \left( \xi _{\infty }\left( J\right) \right) ^{2}\} \) where \(\xi _{\infty }\) is the limiting process of Theorem 1. Proposition 3 (and e.g. [47, Theorem 4.2]) implies that \(\lim _{N\rightarrow \infty } \mathbb {E}\left\{ \xi _{N}\left( J\right) \right\} = \mathbb {E}\left\{ \xi _{\infty }\left( J\right) \right\} \). Therefore, for such \(J_N=J\), it in fact follows from the convergence to Poisson process of Theorem 1 that (1.8) below holds with lim instead of limsup and with equality. We note that the proof of Theorem 1 only requires the upper bound stated in Proposition 4.

  3. 3.

    As we shall see, the only critical points that will be relevant are minimum points.

  4. 4.

    See [47, Theorem 4.2], and note that \(A_{n}\) are continuity sets of \(\bar{\xi }_{\infty }\).

  5. 5.

    This set almost surely contains exactly half of the points in \(\mathscr {C}_{N}\left( L\right) \), and by definition does not contain antipodal points.

  6. 6.

    Up to the negligible event that there exists \(\varvec{\sigma }\in \mathscr {C}_{N}\left( L\right) \) with \(\left\langle \varvec{\sigma },\mathbf {n}\right\rangle =0\).

  7. 7.

    At least when \(\nabla ^{2}\bar{f}_{\varvec{\sigma }}\left( 0\right) \) is invertible, which is the case when \(\mathcal {A}_{1}\) occurs, and we shall indeed restrict to this event when discussing \(Y_{\varvec{\sigma }}\) below.

  8. 8.

    This can be seen by the following. Letting \(\left\{ \tfrac{\partial }{\partial x_{i}}\right\} _{i=1}^{N-1}\) denote the pushforward of \(\left\{ \tfrac{d}{d x_{i}}\right\} _{i=1}^{N-1}\) by P we have that at the north pole, \(\left\{ \tfrac{\partial }{\partial x_{i}} (\mathbf {n})\right\} _{i=1}^{N-1}\) is an orthonormal frame. For any point \(\varvec{\sigma }\) in a small neighborhood of \(\mathbf {n}\) we can define an orthonormal frame as the parallel transport of \(\left\{ \tfrac{\partial }{\partial x_{i}}(\mathbf {n})\right\} _{i=1}^{N-1}\) along the geodesic connecting \(\mathbf {n}\) and \(\varvec{\sigma }\). This yields a smooth orthonormal frame field on this neighborhood, say \(E_i (\varvec{\sigma })=\sum _{j=1}^{N-1}a_{ij} (\varvec{\sigma })\frac{\partial }{\partial x_{j}}(\varvec{\sigma })\), \(i=1,...,N-1\). Working with the coordinate system P one can verify that at \(x=0\) the Christoffel symbols \(\Gamma _{ij}^k\) are equal to 0, and therefore (see e.g. [43, Eq. (2), p. 53]) the derivatives \(\frac{d}{dx_k}a_{ij}(P(x))\) at \(x=0\) are also equal to 0.

  9. 9.

    We note that the integrand in (7.44) is a continuous Radon-Nikodym derivative (as seen from applying the Kac-Rice formula [2, Theorem 12.1.1] to express the mean number of points as above in a subset of the sphere) and therefore it is independent, at each point \(\varvec{\sigma }\), of the choice of the orthonormal frame field \((E_i(\varvec{\sigma }))_{i=1}^{N-1}\).

  10. 10.

    For the cases \(i=2,3,4\) we have continuity and finite variance conditional on \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\) (and not in general), which is, of course, sufficient since we anyway work under this conditioning and since by continuity if \(g_{1}\left( \varvec{\sigma }_{0}\right) \in B_{1}\) for a particular point \(\varvec{\sigma }_{0}\), then there exists a neighborhood of \(\varvec{\sigma }_{0}\) on the sphere on which \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\).


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Author information

Correspondence to Eliran Subag.

Additional information

E. S. acknowledges the support of the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The work of both authors was partially supported by a US-Israel BSF grant and by a grant from the Israel Science Foundation.


Appendix 1: proof of Lemma 7

First we remark that by conditioning on \(\eta \), for any \(x\in \mathbb {R}\),

$$\begin{aligned} \mathbb {E}\left\{ \eta _{\infty }^{+}\left( (-\infty ,x)\right) \right\} \le \mathbb {E} \Big \{\sum _i v(x-\eta _i)\Big \} = \int _{\mathbb {R}}e^{ay} v(x-y) dy, \end{aligned}$$

which is finite due to our assumption on v(x); therefore \(\eta _{\infty }^{+}\) is locally finite.

Let \(g:\,\mathbb {R}\rightarrow \mathbb {R}\) be an arbitrary compactly supported, non-negative function which will be fixed throughout the proof. Let \(\kappa >0\) be a large enough constant such that the support of g is contained in \(\left[ -\kappa ,\kappa \right] \). Denote the event

$$\begin{aligned} B=B_{L,N,\kappa }:=\left\{ \left. \eta _{N}^{+}\right| _{\left[ -\kappa ,\kappa \right] }=\left. \bar{\eta }_{N,L}\right| _{\left[ -\kappa ,\kappa \right] }\right\} . \end{aligned}$$

Since this is the same event as in (6.1), defining \(\epsilon (L)\) by

$$\begin{aligned} \liminf _{N\rightarrow \infty }\mathbb {P}\left\{ B_{L,N,\kappa }\right\} =1-\epsilon \left( L\right) , \end{aligned}$$

we have that \(\epsilon \left( L\right) \rightarrow 0\). Denote \(\left\langle g,\zeta \right\rangle \triangleq \int gd\zeta \) and let \(\mathcal {L}_{\zeta }\left[ g\right] \triangleq \mathbb {E}\left\{ \exp \left\{ -\left\langle g,\zeta \right\rangle \right\} \right\} \) be the Laplace functional of \(\zeta \). Then

$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathcal {L}_{\eta _{N}^{+}}\left[ g\right] -\mathbb {E}\left\{ \mathbf {1}_{B}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \right|= & {} \limsup _{N\rightarrow \infty }\mathbb {E}\left\{ \mathbf {1}_{B^{c}}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \nonumber \\&\le \epsilon \left( L\right) . \end{aligned}$$

Fix some \(L>0\), let \(\delta >0\), and let \(m_{0}:=m_{0}\left( L,\delta \right) \) be a natural number such that \(\mathbb {P}\left\{ D_{L,N,\delta }\right\} >1-\delta \), for all N, with

$$\begin{aligned} D:=D_{L,N,\delta }\triangleq \left\{ Q_{N,L}\le m_{0}\right\} , \end{aligned}$$

where \(Q_{N,L}=\eta _{N}\left( \left[ -L,L\right] \right) \) as defined in point (1) of the lemma. The existence of \(m_{0}\) is guaranteed by tightness of the convergent sequence \(\eta _{N}\). We have that

$$\begin{aligned} \left| \mathbb {E}\left\{ \mathbf {1}_{B}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \right| \le \delta . \end{aligned}$$

Recall that we assumed in point (1) that \(\left. \eta _{N}\right| _{\left[ -L,L\right] }=\sum _{i=1}^{Q_{N,L}}\delta _{\eta _{N,L,i}}\). With \(\hat{\eta }_{N,L}\triangleq \sum _{i=1}^{Q_{N,L}}\delta _{\eta _{N,L,i}+X_{i}}\),

$$\begin{aligned} \mathbf {1}_{B\cap D}\left| \left\langle g,\eta _{N}^{+}\right\rangle -\left\langle g,\,\hat{\eta }_{N,L}\right\rangle \right| \end{aligned}$$

is stochastically dominated by

$$\begin{aligned} \mathbf {1}_{B\cap D}\sum _{i=1}^{m_{0}}\omega _{g}\left( \left| \bar{X}_{N,i}\left( L\right) -X_{i}\right| \right) , \end{aligned}$$

where \(\omega _{g}\left( t\right) =\sup _{\left| x-y\right| \le t}\left| g\left( x\right) -g\left( y\right) \right| \) is the modulus of continuity of g. Since g is uniformly continuous, \(\omega _{g}\) is continuous. Combining this with point (2) of the lemma implies that (7.42) converges in probability to 0. Therefore, since \(e^{-x}\) is bounded and continuous for \(x\ge 0\),

$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\hat{\eta }_{N,L}\right\rangle \right\} \right\} \right| =0, \end{aligned}$$


$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathcal {L}_{\hat{\eta }_{N,L}}\left[ g\right] \right| <1-\liminf _{N\rightarrow \infty }\mathbb {P}\left\{ B_{L,N,\kappa }\cap D_{L,N,\delta }\right\} . \end{aligned}$$

Combining this with the bounds we have on \(\mathbb {P}\left\{ D_{L,N,\delta }\right\} \) and \(\mathbb {P}\left\{ B_{L,N,\kappa }\right\} \), (7.41), and (7.40) yields, upon letting \(\delta \rightarrow 0\),

$$\begin{aligned} \lim _{L\rightarrow \infty }\limsup _{N\rightarrow \infty }\left| \mathcal {L}_{\eta _{N}^{+}} \left[ g\right] -\mathcal {L}_{\hat{\eta }_{N,L}}\left[ g\right] \right| =0. \end{aligned}$$

Thus, in order to complete the proof it is sufficient to show that for some sequence \(L_{k}>0\) such that \(\lim _{k\rightarrow \infty }L_{k}=\infty \),

$$\begin{aligned} \lim _{k\rightarrow \infty }\lim _{N\rightarrow \infty }\mathcal {L}_{\hat{\eta }_{N,L_{k}}} \left[ g\right] =\mathcal {L}_{\eta _{\infty }^{+}}\left[ g\right] . \end{aligned}$$

By [33, Lemma 11.1.II], for all but a countable set of values \(L>0\), the interval \(\left[ -L,L\right] \) is a stochastic continuity set of \(\eta \). Thus we can choose \(L_{k}\) as above such that, from [47, Theorem 4.2],

$$\begin{aligned} \left. \eta _{N}\right| _{\left[ -L_{k},L_{k}\right] }\mathop {\mathop {\rightarrow }\limits _{N\rightarrow \infty }}\limits ^{d}\left. \eta \right| _{\left[ -L_{k},L_{k}\right] }. \end{aligned}$$

Hence also

$$\begin{aligned} \hat{\eta }_{N,L_{k}}\mathop {\mathop {\rightarrow }\limits _{N\rightarrow \infty }}\limits ^{d} \sum _{i:\eta _{i}\in \left[ -L_{k},L_{k}\right] }\delta _{\eta _{i}+X_{i}}, \end{aligned}$$

from which (7.43) follows. \(\square \)

Appendix 2: proof of Lemma 13

We recall that \(f(\varvec{\sigma })=f_{N}(\varvec{\sigma })\) is the unit variance random field on \(\mathbb {S}^{N-1}\triangleq \left\{ \varvec{\sigma }\in \mathbb {R}^{N}:\,\left\| \varvec{\sigma }\right\| _{2}=1\right\} \) given by \(f\left( \varvec{\sigma }\right) =\frac{1}{\sqrt{N}}H_{N}(\sqrt{N}\varvec{\sigma })\), \((E_i(\varvec{\sigma }))_{i=1}^{N-1}\) is an arbitrary piecewise smooth orthonormal frame field on the sphere (w.r.t the standard Riemannian metric), and

$$\begin{aligned} \nabla f\left( \varvec{\sigma }\right) =\left( E_{i}f\left( \varvec{\sigma }\right) \right) _{i=1}^{N-1},\,\,\nabla ^{2}f\left( \varvec{\sigma }\right) =\left( E_{i}E_{j}f\left( \varvec{\sigma }\right) \right) _{i,j=1}^{N-1}. \end{aligned}$$

Provided that certain regularity conditions hold, the Kac-Rice formula [2, Theorem 12.1.1] expresses the mean number of points \(\varvec{\sigma }_{0}\) on the sphere at which \(\nabla f\left( \varvec{\sigma }_{0}\right) =0\), \(\sqrt{N} f\left( \varvec{\sigma }_{0}\right) \in (m_N-L,m_N+L)\), and \(g(\varvec{\sigma }_0)\in B'\), for some random field \(g(\varvec{\sigma })\). For this one needs to apply [2, Theorem 12.1.1] with f in the statement of [2, Theorem 12.1.1] being equal to ‘our’ \(\nabla f\left( \varvec{\sigma }\right) \), and with \(h(\varvec{\sigma })=(\sqrt{N} f\left( \varvec{\sigma }\right) ,g(\varvec{\sigma }))\) and \(B=(m_N-L,m_N+L)\times B'\). This yields

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g(\varvec{\sigma })\in B' \right\} \right\} \nonumber \\&\quad =\mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}: \nabla f\left( \varvec{\sigma }\right) =0,\,h\left( \varvec{\sigma }\right) \in B \right\} \right\} = \int _{\mathbb {S}^{N-1}}d\mu (\varvec{\sigma })\varphi _{\nabla f \left( \varvec{\sigma }\right) }\left( 0\right) \nonumber \\&\qquad \times \mathbb {E}\left\{ \left| \det \left( \nabla ^{2}f \left( \varvec{\sigma }\right) \right) \right| \mathbf {1}\Big \{\left| \sqrt{N}f \left( \varvec{\sigma }\right) -m_{N}\right| <L,\,g\left( \varvec{\sigma }\right) \in B'\Big \}\,\Bigg |\,\nabla f\left( \varvec{\sigma }\right) =0\right\} ,\nonumber \\ \end{aligned}$$

where \(\mu \) is the standard (Hausdorff) measure on the sphere and \(\varphi _{\nabla f\left( \varvec{\sigma }\right) }\left( 0\right) \) is the Gaussian density of \(\nabla f\left( \varvec{\sigma }\right) \) at 0. Assuming that \((f(\varvec{\sigma }),g(\varvec{\sigma }))\) is a stationary field on the sphere, one can replace \(\varvec{\sigma }\) everywhere in the right-hand side above by \(\mathbf {n}\), remove the integration, and multiply the right-hand side above by \(\omega _{N}\) (see (2.2)), the surface area of the \(N-1\)-dimensional unit sphere.Footnote 9 This yields

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g(\varvec{\sigma })\in B' \right\} \right\} = \omega _N\varphi _{\nabla f(\mathbf {n}) }\left( 0\right) \nonumber \\&\quad \times \mathbb {E}\left\{ \left| \det \left( \nabla ^{2}f(\mathbf {n})\right) \right| \mathbf {1}\Big \{\left| \sqrt{N}f(\mathbf {n})-m_{N}\right| <L,\,g \left( \varvec{\sigma }\right) \in B'\Big \}\,\Bigg |\,\nabla f(\mathbf {n})=0\right\} . \end{aligned}$$

In Lemma 13 we stated one case where (7.12) holds as an equality. This case is equivalent to (7.45) without the restrictions on \(g(\varvec{\sigma })\) (after dividing and multiplying by \(\sqrt{Np(p-1)}\) and \((Np(p-1))^\frac{N-1}{2}\) inside the determinant and outside of it, respectively). Thus in order to prove this case what remains is only to check the aforementioned regularity conditions. Those are not difficult to verify when we do not need to worry about the dependence of \(g(\varvec{\sigma })\) and \(f(\varvec{\sigma })\). In particular, the fact that the conditions hold here follows from the case we treat below where we show that they also hold with some \(g(\varvec{\sigma })\) without removing the restrictions on it.

In the other two cases in Lemma 13, (7.12) needs to proved in its original from, with an inequality, for \(2\le i\le 8\); and in a modified form without the condition \(g_1(\varvec{\sigma })\in B_1\) in both sides, for \(i=1\). The proof of the second of the latter two cases is similar to that of the first and therefore we only treat the first. This case is equivalent to (7.12) if \(g(\varvec{\sigma })=(g_i(\varvec{\sigma }),g_1(\varvec{\sigma }))\) and \(B'=B_i^c\times B_1\), and the equality is replaced with an inequality. In order to apply the above to (7.12) directly we will need to verify the regularity conditions and in this case this is not necessarily an easy task. Instead of doing so we relate (7.12) to a similar formula which holds for some modified random fields for which checking the regularity conditions is much easier.

With \(B_{i}^{c}\) denoting the complement of \(B_{i}\) in the corresponding Euclidean space (i.e., in \(\mathbb {R}\) for \(i>2\), \(\mathbb {R}^{\left( N-1\right) ^{2}}\) for \(i=1\), and \(\mathbb {R}^{N-1}\) for \(i=2\)), let \((B_{i}^{c})_\epsilon \) denote the set of points with distance at most \(\epsilon \) from \(B_{i}^{c}\). Let \(Z_{i}\) be a (\(\varvec{\sigma }\)-independent) Gaussian vector in the corresponding Euclidean space all of whose entries are i.i.d standard Gaussian variables. Setting

$$\begin{aligned} h_{i}\left( \varvec{\sigma }\right)&=\left( g_{i}\left( \varvec{\sigma }\right) ,g_{1}\left( \varvec{\sigma }\right) , \sqrt{N}f\left( \varvec{\sigma }\right) \right) ,\\ h_{i,\epsilon }\left( \varvec{\sigma }\right)&=\left( g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i},g_{1}\left( \varvec{\sigma }\right) ,\sqrt{N}f \left( \varvec{\sigma }\right) \right) , \end{aligned}$$

we have that, with \(D_{N}=\left( m_{N}-L,\,m_{N}+L\right) \),

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g_{i}\left( \varvec{\sigma }\right) \notin B_{i},\,g_{1} \left( \varvec{\sigma }\right) \in B_{1}\right\} \right\} \\&\quad =\mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}:\nabla f\left( \varvec{\sigma }\right) =0,\,h_{i}\left( \varvec{\sigma }\right) \in B_{i}^{c}\times B_{1}\times D_{N}\right\} \right\} \\&\quad \le \mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}: \nabla f\left( \varvec{\sigma }\right) =0,\,h_{i,\epsilon _{2}}\left( \varvec{\sigma }\right) \in (B_{i}^{c})_{\epsilon _1}\times B_{1}\times D_{N}\right\} \right\} +\delta \left( \epsilon _{1},\epsilon _{2}\right) , \end{aligned}$$

for \(\epsilon _1,\epsilon _2>0\) and an appropriate function \(\delta (\epsilon _1,\epsilon _2)\) satisfying \(\lim _{\epsilon _{2}\rightarrow 0}\delta \left( \epsilon _{1},\epsilon _{2}\right) =0\).

Thus, provided that we can apply (7.45) in the current situation, we have that

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g_{i}\left( \varvec{\sigma }\right) \notin B_{i},\,g_{1} \left( \varvec{\sigma }\right) \in B_{1}\right\} \right\} \\&\quad \le \delta \left( \epsilon _{1},\epsilon _{2}\right) + \omega _{N}\left( Np \left( p-1\right) \right) ^{\frac{N-1}{2}}\varphi _{\nabla f(\mathbf {n})}\left( 0\right) \mathbb {E}\left\{ \left| \det \left( \frac{\nabla ^{2}f(\mathbf {n})}{\sqrt{Np \left( p-1\right) }}\right) \right| \right. \cdots \\&\quad \left. \mathbf {1}\Big \{\left| \sqrt{N}f(\mathbf {n})-m_{N}\right| <L,\,g_{i} \left( \varvec{\sigma }\right) +\epsilon _{2}Z_{i}\in (B_{i}^{c})_{\epsilon _{1}}, \,g_{1}\left( \varvec{\sigma }\right) \in B_{1}\Big \}\,\Bigg |\,\nabla f(\mathbf {n}) =0\right\} . \end{aligned}$$

Since \(g_{i}\left( \varvec{\sigma }\right) +\epsilon _{2}Z_{i}\mathop {\rightarrow }\limits ^{d}g_{i}\left( \varvec{\sigma }\right) \), as \(\epsilon _{2}\rightarrow 0\), and the indicator function of \((B_{i}^{c})_{\epsilon _{1}}\) is upper semi-continuous, the limit of the expectation in the left-hand side above, as \(\epsilon _{2}\rightarrow 0\), is bounded from above by the same expectation with \(\epsilon _{2}=0\). Therefore, by first taking \(\epsilon _{2}\rightarrow 0\) and then taking \(\epsilon _{1}\rightarrow 0\) (and using the monotone convergence theorem and the fact that \(B_i^c\) is closed) the lemma follows.

All that remains is to verify the conditions required for the application of [2, Theorem 12.1.1], i.e., conditions (a)–(g) there, with f, \(\nabla f\) and h of [2, Theorem 12.1.1] being equal to ‘our’ \(\nabla f\left( \varvec{\sigma }\right) \), \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) and \(h_{i,\epsilon }\left( \varvec{\sigma }\right) \). In particular, we need to account for 8 different cases (of i).

From the definition of the Hamiltonian (1.1), the fact that \(f\left( \varvec{\sigma }\right) \) is Gaussian, and from the Borell-TIS inequality [2, Theorem 2.1.1], the components of \(\nabla f\left( \varvec{\sigma }\right) \), \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) and \(h_{i,\epsilon }\left( \varvec{\sigma }\right) \) (in any of the cases above) are continuous and have finite variance at any \(\varvec{\sigma }\).Footnote 10 Combining this with Corollary 18 which assures the non-degeneracy of the Gaussian variable \(\left( f\left( \varvec{\sigma }\right) ,\nabla f\left( \varvec{\sigma }\right) ,\nabla ^{2}f\left( \varvec{\sigma }\right) \right) \), up to symmetry of the Hessian, conditions (a)–(e) can be verified for all the cases. Condition (f) follows since \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) is Gaussian and stationary. Condition (g) which involves the modulus of continuity of the random fields, is verified for \(f\left( \varvec{\sigma }\right) \), \(\nabla f\left( \varvec{\sigma }\right) \) and \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) directly from the definition of the Hamiltonian (1.1).

All that we have left to show is that condition (g) holds for the components of \(g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i}\) for any \(1\le i\le 8\) and \(\epsilon >0\). Since \(Z_{i}\) does not depend on \(\varvec{\sigma }\), the components of \(g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i}\) and \(g_{i}\left( \varvec{\sigma }\right) \) have the same modulus of continuity, and it is therefore enough to prove that condition (g) holds for the components of \(g_{i}\left( \varvec{\sigma }\right) \). For \(i=1\) this is already proven, since \(g_{1}\left( \varvec{\sigma }\right) \) is equal to \(\nabla ^{2}f\left( \varvec{\sigma }\right) \). For \(i=2,3,4\) (working under the conditioning \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\)), the modulus of continuity of \(g_{i}\left( \varvec{\sigma }\right) \) can be related to that of the components of \(\left( f\left( \varvec{\sigma }\right) ,\nabla f\left( \varvec{\sigma }\right) ,\nabla ^{2}f\left( \varvec{\sigma }\right) \right) \) to prove condition (g).

In order to deal with the case \(i=5,...,8\), we first note that the modulus of continuity of the supremums in the definition of \(g_{i}\) is bounded by that of the functions the supremum of which is taken. The latter is bounded, up to a constant depending on N, by the sum of moduli of continuity of the components of \(f\left( \varvec{\sigma }\right) \), \(\nabla f\left( \varvec{\sigma }\right) \) and \(\nabla ^{2}f\left( \varvec{\sigma }\right) \). From this condition (g) follows in those cases too and the proof is completed. \(\square \)

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Subag, E., Zeitouni, O. The extremal process of critical points of the pure p-spin spherical spin glass model. Probab. Theory Relat. Fields 168, 773–820 (2017). https://doi.org/10.1007/s00440-016-0724-2

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  • Extremal processes
  • Critical points
  • Spin glasses
  • Random fields
  • Random matrices

Mathematics Subject Classification

  • 15A52
  • 82D30
  • 60G60
  • 60G70