The geometry of the Gibbs measure of pure spherical spin glasses

Abstract

We analyze the statics for pure p-spin spherical spin glass models with $p\ge 3$, at low enough temperature. With $F_{N,\beta }$ denoting the free energy, we compute the (logarithmic) second order term of $NF _{N,\beta }$ and prove that, for an appropriate centering $c_{N,\beta },\, NF _{N,\beta }-c_{N,\beta }$ is a tight sequence. We further establish the absence of temperature chaos. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical ‘bands’ centered at deep minima, playing the role of ‘pure states’. For the pure models, the latter makes precise the picture of ‘many valleys separated by high mountains’ and significant parts of the TAP analysis from the physics literature.

Appendix: The Kac–Rice formula and related auxiliary results

The Kac–Rice formula [1, Theorem 12.1.1] is a basic tool in our analysis, allowing us to relate several quantities of interest to the conditional probability $\mathbb {P}_{u,0}$. Below we prove a number of simple derivatives of the formula; some using the results of Sect. 2.2 on critical points and values.

The Kac–Rice formula can be used to express the mean number of critical points $\varvec{\sigma }_{0}$ on the sphere at which the values of some other fields belong to some target set. For us, those other fields will be usually related to masses of bands around $\varvec{\sigma }_{0}$. The variant of the Kac–Rice formula that we use is [1, Theorem 12.1.1]. In the notation of [1, Theorem 12.1.1] we will consider situations where, with some function $g_{N}\left( \varvec{\sigma }\right)$,

$$\begin{aligned} M=\mathbb {S}^{N-1},\,\,\,f\left( \varvec{\sigma }\right) =\nabla H_{N}\left( \varvec{\sigma }\right) ,\,\,\,u=0\in \mathbb {R}^{N-1},\,\,\,h\left( \varvec{\sigma }\right) =\left( H_{N}\left( \varvec{\sigma }\right) ,g_{N}\left( \varvec{\sigma }\right) \right) , \end{aligned}$$

(171)

where we recall that, with $E=(E_{i})_{i=1}^{N-1}$ being an orthonormal frame field on the sphere, we denote

$$\begin{aligned} \nabla H_{N}\left( \hat{\mathbf {n}}\right) =\left( E_{i}H_{N}\left( \hat{\mathbf {n}}\right) \right) _{i=1}^{N-1}\,\,\,\,\,\hbox { and}\,\,\nabla ^{2}H_{N}\left( \hat{\mathbf {n}}\right) =\left( E_{i}E_{j}H_{N}\left( \hat{\mathbf {n}}\right) \right) _{i,j=1}^{N-1}. \end{aligned}$$

The application of [1, Theorem 12.1.1] requires $h\left( \varvec{\sigma }\right)$ to satisfy certain non-degeneracy conditions—namely, conditions (a)–(g) in [1, Theorem 12.1.1]. We will say that $g_{N}\left( \varvec{\sigma }\right)$ is tame if the conditions are satisfied and if $\{h\left( \varvec{\sigma }\right) \}_{\varvec{\sigma }}$ is a stationary random field. Using (1), the conditions are easy to check in any case we will apply the formula and this will be left to the reader. Let

$$\begin{aligned} \omega _{N}=\frac{2\pi ^{N/2}}{\varGamma \left( N/2\right) } \end{aligned}$$

(172)

denote the surface area of the $N-1$-dimensional unit sphere. The following is obtained from a direct application of the formula.

Lemma 13

Suppose that $g_{N}\left( \varvec{\sigma }\right)$ is tame and $D_{N}$ and $J_{N}$ are some intervals, then

$$\begin{aligned}&\mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) :\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| =\omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}} \\&\int _{J_{N}}du\frac{e^{-\frac{u^{2}}{2N}}}{\sqrt{2\pi N}}\mathbb {E}_{u,0}\left\{ \left| \det \left( \frac{\nabla ^{2}H_{N}\left( \hat{\mathbf {n}}\right) }{\sqrt{p\left( p-1\right) \left( N-1\right) /N}}\right) \right| \mathbf {1}\Big \{ g_{N}(\hat{\mathbf {n}})\in D_{N}\Big \}\right\} .\nonumber \end{aligned}$$

(173)

Proof

Applying [1, Theorem 12.1.1] with (171) yields the integral formula

$$\begin{aligned}&\mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) {:}\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| =\int _{\mathbb {S}^{N-1}}d\nu \left( \varvec{\sigma }\right) \phi _{\nabla H_{N}\left( \varvec{\sigma }\right) }\left( 0\right) \\&\quad \times \,\mathbb {E}\left\{ \left. \left| \det \left( \nabla ^{2}H_{N}\left( \varvec{\sigma }\right) \right) \right| \mathbf {1}\Big \{ H_{N}\left( \varvec{\sigma }\right) \in J_{N},\,g_{N}\left( \varvec{\sigma }\right) \in D_{N}\Big \}\,\right| \,\nabla H_{N}\left( \varvec{\sigma }\right) =0\right\} ,\nonumber \end{aligned}$$

(174)

where $\phi _{\nabla H_{N}\left( \varvec{\sigma }\right) }\left( x\right)$ is the Gaussian density of $\nabla H_{N}\left( \varvec{\sigma }\right)$ and $\nu$ is the standard measure on the sphere (not normalized). Since the integrand above is independent of $\varvec{\sigma }$,¹⁰ we can replace the integral with the value of the integrand evaluated at $\varvec{\sigma }=\hat{\mathbf {n}}$ and multiply by $\omega _{N}N^{\frac{N-1}{2}}$, the volume of $\mathbb {S}^{N-1}$. By Lemma 2, $\nabla H_{N}\left( \hat{\mathbf {n}}\right) \sim \mathcal {N}\left( 0,p\mathbf {I}_{N-1}\right)$, so that $\phi _{\nabla H_{N}\left( \varvec{\sigma }\right) }\left( 0\right) =(2\pi p)^{-\frac{N-1}{2}}$, and $H_{N}\left( \hat{\mathbf {n}}\right)$ and $\nabla H_{N}\left( \hat{\mathbf {n}}\right)$ are independent. The proof is completed by conditioning on $H_{N}\left( \hat{\mathbf {n}}\right)$ in addition to $\nabla H_{N}\left( \varvec{\sigma }\right) =0$ and some calculus. $\square$

Several derivatives of (173) are of use to us. Their proofs will involve the rate function $\varTheta _{p}\left( E\right)$ of Theorem 5 which we now define explicitly. Let $\nu ^{*}$ denote the semicircle measure, the density of which with respect to Lebesgue measure is

$$\begin{aligned} \frac{d\nu ^{*}}{dx}=\frac{1}{2\pi }\sqrt{4-x^{2}}\mathbf {1}_{\left| x\right| \le 2}, \end{aligned}$$

(175)

and define the function (see e.g., [23, Proposition II.1.2])

$$\begin{aligned} \varOmega (x)&\triangleq \int _{\mathbb {R}}\log \left| \lambda -x\right| d\nu ^{*}\left( \lambda \right) \\&={\left\{ \begin{array}{ll} \frac{x^{2}}{4}-\frac{1}{2}, &{}\quad \hbox { if }0\le \left| x\right| \le 2,\\ \frac{x^{2}}{4}-\frac{1}{2}-\left[ \frac{\left| x\right| }{4}\sqrt{x^{2}-4}-\log \left( \sqrt{\frac{x^{2}}{4}-1}+\frac{\left| x\right| }{2}\right) \right] , &{}\quad \hbox { if }\left| x\right| >2. \end{array}\right. }\nonumber \end{aligned}$$

(176)

The exponential growth rate function of (15) is given [5] by

$$\begin{aligned} \varTheta _{p}\left( E\right) ={\left\{ \begin{array}{ll} \frac{1}{2}+\frac{1}{2}\log \left( p-1\right) -\frac{E^{2}}{2}+\varOmega \left( \gamma _{p}E\right) , &{}\quad \hbox { if }u<0,\\ \frac{1}{2}\log \left( p-1\right) , &{}\quad \hbox { if }u\ge 0, \end{array}\right. } \end{aligned}$$

(177)

where $\gamma _{p}=\sqrt{p/\left( p-1\right) }$.

Lemma 14

Suppose that $g_{N}\left( \varvec{\sigma }\right)$ is tame, $D_{N}$ are some intervals and $J_{N}= NJ$ where $J\subset \left( -\infty ,0\right)$ is a fixed finite interval. Let $\varphi {:}\,\mathbb {R}\rightarrow \mathbb {R}$ be a continuous function and assume that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\left\{ \frac{1}{N}\log \left( \mathbb {P}_{u,0}\left\{ g_{N}\left( \hat{\mathbf {n}}\right) \in D_{N}\right\} \right) -\varphi \left( \frac{u}{N}\right) \right\} \le 0. \end{aligned}$$

(178)

Then

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N}\log \left( \mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) {:}\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| \right) \le \sup _{E\in J}\left\{ \varTheta _{p}\left( E\right) +\varphi \left( E\right) \right\} . \end{aligned}$$

(179)

Proof

From (173) and Hölder’s inequality,

$$\begin{aligned}&\mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) {:}\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| \le \omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}\\&\quad \times \int _{J_{N}}du\frac{1}{\sqrt{2\pi N}}e^{-\frac{u^{2}}{2N}}\left( \mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| ^{a}\right\} \right) ^{1/a}\left( \mathbb {P}_{u,0}\left\{ g_{N}\left( \hat{\mathbf {n}}\right) \in D_{N}\right\} \right) ^{1/b},\nonumber \end{aligned}$$

(180)

where

$$\begin{aligned} \mathbf {V}\triangleq \frac{\nabla ^{2}H_{N}\left( \hat{\mathbf {n}}\right) }{\sqrt{p\left( p-1\right) \left( N-1\right) /N}}, \end{aligned}$$

(181)

$a>1$ is an arbitrary number, and $b:=b(a)=a/\left( a-1\right)$.

First,

$$\begin{aligned} \lim _{N\rightarrow \infty }\omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}=\frac{1}{2}+\frac{1}{2}\log \left( p-1\right) . \end{aligned}$$

(182)

Second, by a change of variables $u\mapsto Nv$,

$$\begin{aligned}&\limsup _{N\rightarrow \infty }\frac{1}{N}\log \left( \int _{J_{N}}du\frac{1}{\sqrt{2\pi N}}e^{-\frac{u^{2}}{2N}}\exp \left\{ N\left( \varOmega \left( \gamma _{p}\frac{u}{N}\right) +\varphi \left( \frac{u}{N}\right) \right) \right\} \right) \nonumber \\&\quad \le \sup _{x\in J}\left\{ \varOmega \left( \gamma _{p}x\right) -\frac{x^{2}}{2}+\varphi \left( x\right) \right\} , \end{aligned}$$

(183)

where $\gamma _{p}=\sqrt{p/\left( p-1\right) }$ and $\varOmega$ is defined in (176). Thus, by (177), the lemma follows if we show that for any $a>1$,

$$\begin{aligned} \limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\left\{ \frac{1}{Na}\log \left( \mathbb {E}_{u,0}\left\{ \left| \det \mathbf {V}\right| ^{a}\right\} \right) -\varOmega \left( \gamma _{p}\frac{u}{N}\right) \right\} \le 0, \end{aligned}$$

(184)

and that

$$\begin{aligned} \limsup _{a\rightarrow \infty }\limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\left\{ \frac{1}{Nb(a)}\log \left( \left( \mathbb {P}_{u,0}\left\{ g_{N}\left( \hat{\mathbf {n}}\right) \in D_{N}\right\} \right) \right) -\varphi \left( \frac{u}{N}\right) \right\} \le 0. \end{aligned}$$

(185)

The inequality (185) obviously follows from (178), since $\lim _{a\rightarrow \infty }b(a)=1$.

From (38) and (42), the conditional law of $\mathbf {V}$ under $\mathbb {P}_{u,0}\left\{ \cdot \right\}$ is identical to that of

$$\begin{aligned} \mathbf {M}_{u}:=\mathbf {M}_{u,N-1}\triangleq \mathbf {M}-\gamma _{p}\frac{u}{\sqrt{N\left( N-1\right) }}\mathbf {I}, \end{aligned}$$

(186)

where $\mathbf {M}$ is a GOE matrix and $\mathbf {I}$ is the identity matrix, both of dimension $N-1$. For any $0<\epsilon<1<\kappa$, with $\lambda _{i}$ denoting the eigenvalues of $\mathbf {M}$ and $\lambda _{*}\left( u\right) =\max _{i}\left| \lambda _{i}-\gamma _{p}\frac{u}{\sqrt{N\left( N-1\right) }}\right|$,

$$\begin{aligned} \mathbb {E}\left\{ \left| \det \left( \mathbf {M}_{u}\right) \right| ^{a}\right\}&\le \mathbb {E}\left\{ \exp \left\{ a\sum _{i}\log _{\epsilon }^{\kappa }\left( \left| \lambda _{i}-\gamma _{p}\frac{u}{\sqrt{N\left( N-1\right) }}\right| \right) \right\} \right\} \nonumber \\&\quad +\,\mathbb {E}\left\{ \left( \lambda _{*}\left( u\right) \right) ^{a\left( N-1\right) }\mathbf {1}\left\{ \left( \lambda _{*}\left( u\right) \right) \ge \kappa \right\} \right\} , \end{aligned}$$

(187)

where

$$\begin{aligned} \log _{\epsilon }^{\kappa }\left( x\right)&={\left\{ \begin{array}{ll} \log \left( \epsilon \right) &{}\quad \,\,\hbox {if }x\le \epsilon ,\\ \log x &{}\quad \,\,\hbox {if }x\in \left( \epsilon ,\kappa \right) ,\\ \log \kappa &{}\quad \,\,\hbox {if }x\ge \kappa . \end{array}\right. } \end{aligned}$$

(188)

From the upper bound on the maximal eigenvalue of [8, Lemma 6.3],

$$\begin{aligned} \lim _{\kappa \rightarrow \infty }\limsup _{N\rightarrow \infty }\frac{1}{N}\log \left( \mathbb {E}\left\{ \left( \lambda _{*}\left( u\right) \right) ^{a\left( N-1\right) }\mathbf {1}\left\{ \left( \lambda _{*}\left( u\right) \right) \ge \kappa \right\} \right\} \right) =-\infty , \end{aligned}$$

(189)

uniformly for $u\in J_{N}$. The empirical measure of eigenvalues of GOE matrices $L_{N}=\frac{1}{N-1}\sum _{i=1}^{N-1}\lambda _{i}$ satisfies a large deviation principle with speed $N^{2}$ and a good rate function $J_{0}\left( \nu \right)$, in the space of Borel probability measures on $\mathbb {R}$ equipped with the weak topology, which is compatible with the Lipschitz bounded metric; see [9, Theorem 2.1.1]. The good rate function $J_{0}\left( \nu \right)$ satisfies $J_{0}\left( \nu \right) =0$ if and only if $\nu =\nu ^{*}$ is the semicircle law (175). Combined with the fact that the functions $\log _{\epsilon }^{\kappa }\left( \left| \cdot -u'\right| \right)$ have the same Lipschitz constant and bound for all $u'\in \mathbb {R}$, this implies that for the event

$$\begin{aligned}&F\left( u,\delta \right) \\&\quad =\left\{ \left| \frac{1}{N-1}\sum _{i}\log _{\epsilon }^{\kappa }\left| \lambda _{i}-\gamma _{p}\frac{u}{\sqrt{N\left( N-1\right) }}\right| -\int \log _{\epsilon }^{\kappa }\left| \lambda -\gamma _{p}\frac{u}{N}\right| d\mu ^{*}\left( \lambda \right) \right| >\delta \right\} , \end{aligned}$$

we have for any $\delta >0$ some positive number $d\left( \delta \right)$ such that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\frac{1}{N^{2}}\log \left( \mathbb {P}\left\{ F\left( u,\delta \right) \right\} \right) <-d\left( \delta \right) . \end{aligned}$$

By taking $\kappa$ large enough and $\epsilon$ small enough, combining this with (187) and (189) proves (184) and completes the proof. $\square$

For intervals $J_{N}$ of length $o\left( N\right)$ around $-NE_{0}$ the following is also useful for us.

Lemma 15

Suppose that $g_{N}\left( \varvec{\sigma }\right)$ is tame, $D_{N}$ are some intervals and $J_{N}=\left( m_{N}-a_{N},m_{N}+a_{N}^{\prime }\right)$ for some sequences $a_{N},\,a_{N}^{\prime }=o\left( N\right)$. Let $c_{p}$ be as defined in (18). Then for large enough N,

$$\begin{aligned}&\mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) {:}\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| \nonumber \\&\quad \le C\int _{J_{N}}du\cdot e^{c_{p}\left( u-m_{N}\right) }\left( \mathbb {P}_{u,0}\left\{ g_{N}\left( \hat{\mathbf {n}}\right) \in D_{N}\right\} \right) ^{1/2}, \end{aligned}$$

(190)

where $C>0$ is an appropriate constant.

Proof

We shall use the notation (181) introduced in the proof of Lemma 14. Since the conditional law of $\mathbf {V}$ under $\mathbb {P}_{u,0}\left\{ \cdot \right\}$ is identical to that of the shifted GOE matrix (186), as a particular case of [34, Corollary 23],

$$\begin{aligned} \left( \mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| ^{2}\right\} \right) ^{1/2}\le C\mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| \right\} , \end{aligned}$$

(191)

uniformly in $u\in J_{N}$, for some constant $C>0$. As in the proof of Lemma 14, (180) holds and therefore, taking $a=2$ and using 191, we obtain that

$$\begin{aligned}&\mathbb {E}\left| \left\{ \varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) :\,g_{N}\left( \varvec{\sigma }_{0}\right) \in D_{N}\right\} \right| \le C\omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}\\&\quad \int _{J_{N}}du\frac{1}{\sqrt{2\pi N}}e^{-\frac{u^{2}}{2N}}\mathbb {E}_{u,0}\left\{ \left| \det (\mathbf {V})\right| \right\} \left( \mathbb {P}_{u,0}\left\{ g_{N}\left( \hat{\mathbf {n}}\right) \in D_{N}\right\} \right) ^{1/2},\nonumber \end{aligned}$$

(192)

By Lemma 13,

$$\begin{aligned} \mathbb {E}\left| \mathscr {C}_{N}\left( J_{N}\right) \right| =\int _{J_{N}}du\omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}\frac{1}{\sqrt{2\pi N}}e^{-\frac{u^{2}}{2N}}\mathbb {E}_{u,0}\left\{ \left| \det (\mathbf {V})\right| \right\} . \end{aligned}$$

By definition, up to a factor of 2 in the even p case [related to the normalization factor in (19)], the integrand above is equal to the density of the intensity measure of the extremal point process of critical points $\xi _{N}$, defined in (19), shifted by $m_{N}$. Thus, by [35, Proposition 3] it converges uniformly to $e^{c_{p}\left( u-m_{N}\right) }$ (see also Theorem 7 above). Combined with (192) this yields (190). $\square$

The non-negative random variable

$$\begin{aligned} \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q,q^{\prime }\right) \right) \end{aligned}$$

can be approximated by

$$\begin{aligned} \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }\sum _{i\le k}t_{i}\mathbf {1}\left\{ Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q,q^{\prime }\right) \right) \in \left[ t_{i},t_{i+1}\right) \right\} , \end{aligned}$$

where $\left[ t_{i},t_{i+1}\right) ,\,i\le k$, form a finite partition of $\left[ 0,\infty \right)$. Combining this with the monotone convergence theorem and (173) one obtains the following.

Corollary 8

We have that

$$\begin{aligned}&\mathbb {E}\left\{ \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q_{N},q_{N}^{\prime }\right) \right) \right\} =\omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}\\&\quad \int _{J_{N}}du\frac{e^{-\frac{u^{2}}{2N}}}{\sqrt{2\pi N}}\mathbb {E}_{u,0}\left\{ \left| \det \frac{\nabla ^{2}H_{N}\left( \hat{\mathbf {n}}\right) }{\sqrt{p\left( p-1\right) \left( N-1\right) /N}}\right| Z_{N,\beta }(\mathrm{Band}(\hat{\mathbf {n}},q_{N},q_{N}^{\prime }))\right\} .\nonumber \end{aligned}$$

(193)

We have the following exponential bound on the expectation above.

Lemma 16

Let $J_{N}= NJ$ where $J\subset \mathbb {R}$ is a fixed interval. Let $\varphi {:}\,\mathbb {R}\rightarrow \mathbb {R}$ be a continuous function and suppose that

$$\begin{aligned} \limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\left\{ \frac{1}{N}\log \left( \mathbb {E}_{u,0}\left\{ Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}},q_{N},q_{N}^{\prime }\right) \right) \right\} \right) -\varphi \left( \frac{u}{N}\right) \right\} \le 0. \end{aligned}$$

(194)

Then

$$\begin{aligned}&\limsup _{N\rightarrow \infty }\frac{1}{N}\log \Bigg (\mathbb {E}\Bigg \{ \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q_{N},q_{N}^{\prime }\right) \right) \Bigg \} \Bigg )\\&\quad \le \sup _{x\in J}\left\{ \varTheta _{p}\left( x\right) +\varphi \left( x\right) \right\} ,\nonumber \end{aligned}$$

(195)

Proof

Abbreviate $\mathrm{Band}\left( \varvec{\sigma }_{0}\right) =\mathrm{Band}\left( \varvec{\sigma }_{0},q_{N},q_{N}^{\prime }\right)$. By Hölder’s inequality,

$$\begin{aligned} b\mapsto \log \left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right) ^{b}\right\} \right) \end{aligned}$$

is a convex function. Hence, for $b\in \left( 1,2\right)$, for any $u\in J_{N}$,

$$\begin{aligned}&\frac{1}{Nb}\log \left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right) ^{b}\right\} \right) \nonumber \\&\quad \le \frac{1}{Nb}\left( \left( 2-b\right) \log \left( \mathbb {E}_{u,0}\left\{ Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right\} \right) +\left( b-1\right) N\bar{C}\right) , \end{aligned}$$

(196)

where we define

$$\begin{aligned} \bar{C}:=2\beta ^{2}+\frac{2\beta }{N}\sup _{u\in J_{N}}|u|=2\beta ^{2}+\frac{2\beta }{N}\sup _{x\in J}|x| \end{aligned}$$

and use the fact that

$$\begin{aligned}&\log \left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right) ^{2}\right\} \right) \\&\quad \le \log \left( \max _{\varvec{\sigma },\varvec{\sigma }'\in \mathbb {S}^{N-1}}\mathbb {E}_{u,0}\left\{ e^{-\beta H_{N}\left( \varvec{\sigma }\right) -\beta H_{N}\left( \varvec{\sigma }'\right) }\right\} \right) \le 2\beta |u|+2\beta ^{2}N. \end{aligned}$$

For any $b>1$, with $a:=a(b)=b/\left( b-1\right)$, we have, using the notation (181),

$$\begin{aligned}&\mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right\} \\&\quad \le \left( \mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| ^{a}\right\} \right) ^{1/a}\left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right) ^{b}\right\} \right) ^{1/b}.\nonumber \end{aligned}$$

(197)

Using (196) and (184) and letting $b\searrow 1$ we obtain

$$\begin{aligned}&\limsup _{N\rightarrow \infty }\sup _{u\in J_{N}}\Bigg \{\frac{1}{N}\log \left( \mathbb {E}_{u,0}\left\{ \left| \det \left( \mathbf {V}\right) \right| Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}}\right) \right) \right\} \right) \\&\qquad -\varOmega \left( \gamma _{p}\frac{u}{N}\right) -\varphi \left( \frac{u}{N}\right) \Bigg \} \le 0. \end{aligned}$$

From (193), (182) and (183), we conclude that (195) follows. $\square$

We finish with the following lemma

Lemma 17

Let $J_{N}=\left( m_{N}-a_{N},m_{N}+a_{N}^{\prime }\right)$ with some sequences $a_{N},\,a_{N}^{\prime }=o\left( N\right)$, and define $c_{p}$ as in (18). Then, for large enough N,

$$\begin{aligned}&\mathbb {E}\left\{ \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q_{N},q_{N}^{\prime }\right) \right) \right\} \\&\quad \le C\cdot \int _{J_{N}}du\cdot e^{c_{p}\left( u-m_{N}\right) }\left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}},q_{N},q_{N}^{\prime }\right) \right) \right) ^{2}\right\} \right) ^{1/2},\nonumber \end{aligned}$$

(198)

where $C>0$ is an appropriate constant.

Proof

By (193), (197) and (191) it follows that, for an appropriate $C>0$, for large N,

$$\begin{aligned}&\mathbb {E}\left\{ \sum _{\varvec{\sigma }_{0}\in \mathscr {C}_{N}\left( J_{N}\right) }Z_{N,\beta }\left( \mathrm{Band}\left( \varvec{\sigma }_{0},q_{N},q_{N}^{\prime }\right) \right) \right\} \le C\cdot \omega _{N}\left( \left( N-1\right) \frac{p-1}{2\pi }\right) ^{\frac{N-1}{2}}\\&\int _{J_{N}}du\frac{1}{\sqrt{2\pi N}}e^{-\frac{u^{2}}{2N}}\mathbb {E}_{u,0}\left\{ \left| \det (\mathbf {V})\right| \right\} \left( \mathbb {E}_{u,0}\left\{ \left( Z_{N,\beta }\left( \mathrm{Band}\left( \hat{\mathbf {n}},q_{N},q_{N}^{\prime }\right) \right) \right) ^{2}\right\} \right) ^{1/2}, \end{aligned}$$

where $\mathbf {V}$ is defined in (181). The proof now follows from the argument used in the proof of Lemma 15 right after (192). $\square$