Given $\varsigma\in\mathbb{R}^{N}$, we define

$H_{N}^{\text{sym}}(\varsigma)=H_{N}(\varsigma,\dots,\varsigma).$

We let $H_{N}^{\text{sym},1},\dots,H_{N}^{\text{sym},D}$ be $D$ independent
copies of the Gaussian process $H_{N}^{\text{sym}}$ and we assume that
$(H_{N}^{\text{sym},1},\dots,H_{N}^{\text{sym},D})$ is independent of $H_{N}$.
For every $\sigma=(\sigma_{1},\dots,\sigma_{D})\in\mathbb{R}^{D\times N}$, we
define

(7.7)
$K_{N}(\sigma)=\frac{1}{\sqrt{D}}\sum_{d=1}^{D}H^{\text{sym},d}_{N}(\sigma_{d}).$

The Gaussian process $K_{N}$ satisfies

$\mathbb{E}\left[K_{N}(\sigma)K_{N}(\tau)\right]=N\Xi\left(R(\sigma,\tau)\right),$

where $\Xi(x)=\frac{1}{D}\sum_{d=1}^{D}\xi_{\dagger}\left(x_{d}\right)$. Note
that, $\Xi_{\dagger}$ =$\xi_{\dagger}$, also note that according to
Proposition 7.2, the function $\Xi$ is convex on $\mathbb{R}^{D}_{+}$. In
particular, the family of Gaussian processes $(K_{N})_{N\geqslant 1}$ is
covered by Theorem 1.3. Thus, the free energy of the Gaussian process $K_{N}$
converges when $N\to+\infty$ and the limit free energy can be expressed using
a variational formula. We let $\overline{F}_{N}$ denote the free energy of
$H_{N}$ as in (1.2), and we let $\overline{G}_{N}$ denote the free energy of
$K_{N}$.

######  Theorem 7.6.

Assume that $\xi$ satisfies (7.2), we have for every $t\geqslant 0$,

(7.8)
$\limsup_{N\to+\infty}\overline{F}_{N}(t)\leqslant\lim_{N\to+\infty}\overline{G}_{N}(t).$

######  Remark 7.7.

According to Proposition 7.2, as soon as $\xi$ is permutation-invariant, $\xi$
satisfies (7.2). So we can deduce Theorem 1.4 from Theorem 7.6 by applying
Theorem 1.3 to the Hamiltonian $K_{N}$ and observing that,

$\left(\xi_{\dagger}\left(\frac{\cdot}{D}\right)\right)^{*}(\lambda)=\Xi^{*}(\lambda,\dots,\lambda).$

######  Remark 7.8.

Note that in fact, as stated, neither $P_{1}$ nor $\xi$ are required to be
permutation-invariant for Theorem 7.6 to hold. But if $P_{1}$ or $\xi$ is not
permutation-invariant, there is no reason to believe that (7.8) is a tight
upper bound, and in fact the proof of Theorem 7.10 given below breaks down in
this case.

###### Proof.

Let $H^{\text{pert}}_{N}$ be the Gaussian process defined in (7.4), we take it
independent of $H_{N}$ and $K_{N}$. Recall that the covariance of
$H^{\text{pert}}_{N}$ depends on some parameter $u$ and we denote
$\mathbb{E}_{u}$ the expectation with respect to this parameter. Without loss
of generality, throughout the proof we assume that $t=1/2$. For every
$\lambda\in[0,1]$, define

$\displaystyle H_{N,\lambda}(\sigma)=$
$\displaystyle\left(\sqrt{1-\lambda}H_{N}(\sigma)-(1-\lambda)\frac{N}{2}\xi\left(R(\sigma,\sigma)\right)\right)$
$\displaystyle+\left(\sqrt{\lambda}K_{N}(\sigma)-\lambda\frac{N}{2}\Xi\left(R(\sigma,\sigma)\right)\right)$
$\displaystyle+c_{N}H^{\text{pert}}_{N}(\sigma).$

Where $c_{N}=N^{-\omega}$ with $0<\omega<1/2$. We let
$\langle\cdot\rangle_{\lambda}$ denote the Gibbs measure associated to
$H_{N,\lambda}(\sigma)$ and we also define the associated free energy

$\varphi_{N}(\lambda)=-\frac{1}{N}\mathbb{E}\log\int\exp\left(H_{N,\lambda}(\sigma)\right)\mathrm{d}P_{N}(\sigma)$

Here, as previously, we use the symbol $\mathbb{E}$ to denote expectation with
respect to every source of randomness involved. In particular, $\mathbb{E}$
integrates out the randomness of $u$ and $\varphi_{N}(\lambda)$ is a nonrandom
quantity. Note that crucially, thanks to (7.5), the nature of the term
$c_{N}H^{\text{pert}}_{N}$ is indeed perturbative as it does not influence the
value of the limit free energy of $\langle\cdot\rangle_{\lambda}$. Inequality
(7.8) can be rewritten as

$\limsup_{N\to+\infty}\varphi_{N}(0)\leqslant\limsup_{N\to+\infty}\varphi_{N}(1).$

We will prove this inequality by observing that $\varphi_{N}$ is almost a
nondecreasing function of $\lambda$. For every $\lambda\in(0,1)$,
$\varphi_{N}$ is differentiable at $\lambda$ and we have

$\displaystyle\varphi_{N}^{\prime}(\lambda)$
$\displaystyle=-\frac{1}{N}\mathbb{E}\left\langle\frac{d}{d\lambda}H_{N,\lambda}(\sigma)\right\rangle_{\lambda}$
$\displaystyle=-\frac{1}{N}\left(\mathbb{E}\left\langle-\frac{1}{2\sqrt{1-\lambda}}H_{N}(\sigma)+\frac{1}{2\sqrt{\lambda}}K_{N}(\sigma)+\frac{N}{2}(\xi-\Xi)\left(R(\sigma,\sigma)\right)\right\rangle_{\lambda}\right),$

Let us now treat the first term in the above display using Gaussian
integration by parts [20, Lemma 1.1]. For every
$\sigma,\tau\in\mathbb{R}^{D}$, we have

$\displaystyle\mathbb{E}\left[\left(-\frac{1}{2\sqrt{1-\lambda}}H_{N}(\sigma)+\frac{1}{2\sqrt{\lambda}}K_{N}(\sigma)\right)H_{N,\lambda}(\tau)\right]$
$\displaystyle=\mathbb{E}\left[-\frac{1}{2\sqrt{1-\lambda}}H_{N}(\sigma)\sqrt{1-\lambda}H_{N}(\tau)\right]+\mathbb{E}\left[\frac{1}{2\sqrt{\lambda}}K_{N}(\sigma)\sqrt{\lambda}K_{N}(\tau)\right]$
$\displaystyle+\mathbb{E}[c_{N}H_{N}^{\text{pert}}(\sigma)]\mathbb{E}\left[-\frac{1}{2\sqrt{1-\lambda}}H_{N}(\sigma)+\frac{1}{2\sqrt{\lambda}}K_{N}(\sigma)\right]$
$\displaystyle+\left((1-\lambda)\xi+\lambda\Xi\right)\mathbb{E}\left[-\frac{1}{2\sqrt{1-\lambda}}H_{N}(\sigma)+\frac{1}{2\sqrt{\lambda}}K_{N}(\sigma)\right]$
$\displaystyle=-\frac{1}{2}N\xi\left(R(\sigma,\tau)\right)+\frac{1}{2}N\Xi\left(R(\sigma,\tau)\right)+0+0$
$\displaystyle=\frac{N}{2}(\Xi-\xi)\left(R(\sigma,\tau)\right).$

Thus, the Gaussian integration by parts formula yields

$\varphi^{\prime}_{N}(\lambda)=\frac{1}{2}\mathbb{E}\left\langle(\Xi-\xi)\left(R(\sigma,\tau)\right)\right\rangle.$

From Remark 7.4, if the power series that defines the function $\xi$ does not
include any term of odd degree, then $\Xi-\xi$ is nonnegative on
$\mathbb{R}^{D}$ which implies that $\varphi^{\prime}_{N}(\lambda)\geqslant 0$
and we do not need the perturbation $c_{N}H^{\text{pert}}_{N}(\sigma)$ in this
case. However, if some terms of odd degree are present, then according to
Proposition 7.2 and Remark 7.3, we can only claim that $\Xi-\xi$ is
nonnegative on $\mathbb{R}_{+}^{D}$. Since the overlaps
$\frac{\sigma_{d}\cdot\tau_{d}}{N}$ can take values in $[-1,1]$, the previous
display only implies

$\varphi^{\prime}_{N}(\lambda)\geqslant\delta(\varepsilon)-\frac{\|\Xi-\xi\|_{\infty}}{2}\mathbb{E}\left\langle\mathbf{1}_{\exists
d,\;\frac{\sigma_{d}\cdot\tau_{d}}{N}\leqslant-\varepsilon}\right\rangle_{\lambda},$

where $\delta(\varepsilon)=\min\left\\{\frac{(\Xi-\xi)(R)}{2}\big{|}\;\forall
d,\;R_{d}\in(-\varepsilon,1]\right\\}$. Since $\Xi-\xi$ is locally Lipschitz
and nonnegative on $\mathbb{R}_{+}^{D}$, there exists $C>0$ such that
$\delta(\varepsilon)\geqslant-C\varepsilon$. Therefore,

$\varphi_{N}(1)\geqslant\varphi_{N}(0)-C\varepsilon-\frac{\|\Xi-\xi\|_{\infty}}{2}\int_{0}^{1}\mathbb{E}\left\langle\mathbf{1}_{\exists
d,\;\frac{\sigma_{d}\cdot\tau_{d}}{N}\leqslant-\varepsilon}\right\rangle_{\lambda}\mathrm{d}\lambda.$

To control the remaining term, observe that

$\displaystyle\mathbb{E}\left\langle\mathbf{1}_{\exists
d,\;\frac{\sigma_{d}\cdot\tau_{d}}{N}\leqslant-\varepsilon}\right\rangle_{\lambda}\leqslant\sup_{H}\mathbb{E}_{u}\mathbb{E}_{H_{N}^{\text{pert}}}\widetilde{G}_{N}^{\otimes
2}\left(\exists
d,\;\frac{\sigma_{d}\cdot\tau_{d}}{N}\leqslant-\varepsilon\right),$

where, as in Lemma 7.5, the supremum is taken over all nonrandom Hamiltonians
$H$, $\mathbb{E}_{H_{N}^{\text{pert}}}$ denotes the expectation with respect
to $H_{N}^{\text{pert}}$ and $\widetilde{G}_{N}$ is the Gibbs measure of
$H+c_{N}H_{N}^{\text{pert}}$. In particular, the right-hand side in the
previous display is independent of $\lambda$ and according to Lemma 7.5, it
vanishes in the limit $N\to+\infty$. Letting $N\to+\infty$, we obtain

$\limsup_{N\to+\infty}\varphi_{N}(1)\geqslant\limsup_{N\to+\infty}\varphi_{N}(0)-C\varepsilon-0.$

Since $\varepsilon>0$ is arbitrary, we can conclude by letting $\varepsilon\to
0$. ∎

###  7.4. Connections with the Hamilton-Jacobi approach

When $\xi$ is nonconvex, the Parisi formula completely breaks down. To the
best of our knowledge, until recently, it seems that there was no clear
conjecture on what the limit of the free energy should be in this case. In
[16, Conjecture 2.6], it is proposed that results such as Theorem 1.6 should
generalize to nonconvex models. It was later shown in [17, 15] that the
$\liminf$ of $\overline{F}_{N}(t)$ as $N\to+\infty$ is lower bounded in terms
of the viscosity solution of a Hamilton-Jacobi equation. This lower bound
holds, regardless of convexity and permutation invariance.

Recall that $\xi_{\dagger}(\lambda)=\xi(\lambda,\dots,\lambda)$ and
$\psi^{\dagger}(p)=\psi(p,\dots,p)$, also recall that

$\Xi(x)=\frac{1}{D}\sum_{d=1}^{D}\xi(x_{d},\dots,x_{d}).$

If we combine the lower bound of [17, 15] with Theorem 7.6 and 5.9, we obtain
the following proposition.

######  Proposition 7.9.

Assume that $\xi$ is permutation-invariant and only depends on the diagonal
coefficients of its argument, assume that $P_{1}$ is permutation-invariant.
Then, even when $\xi$ is nonconvex on $S^{D}_{+}$, we have

(7.9)
$f(t,0)\leqslant\liminf_{N\to+\infty}\overline{F}_{N}(t)\leqslant\limsup_{N\to+\infty}\overline{F}_{N}(t)\leqslant
g(t,0),$

where $f$ and $g$ are the viscosity solutions of the following equations,

(7.10) $\begin{cases}\partial_{t}f-\int\xi(\nabla f)=0\text{ on
}(0,\infty)\times(\mathcal{Q}(\mathbb{R}^{D}_{+})\cap L^{2})\\\
f(0,\cdot)=\psi\text{ on }\mathcal{Q}(\mathbb{R}^{D}_{+})\cap
L^{2},\end{cases}$

(7.11) $\begin{cases}\partial_{t}g-\int\xi_{\dagger}\left(\frac{\nabla
g}{D}\right)=0\text{ on }(0,+\infty)\times(\mathcal{Q}(\mathbb{R}_{+})\cap
L^{2})\\\ g(0,\cdot)=\psi^{\dagger}\text{ on }\mathcal{Q}(\mathbb{R}_{+})\cap
L^{2}.\end{cases}$

The results of Section 5.2 can be regarded as a proof of the fact
$f(t,0)=g(t,0)$ when $\xi$ is convex. To prove this statement we only relied
on the convexity of $\xi_{\dagger}$ and the fact that $f$ is not simply a
solution in the viscosity sense but also a Gateaux differentiable solution. To
the best of our knowledge, there is no known proof of the fact that $f$ is
Gateaux differentiable when $\xi$ is nonconvex. Indeed, in the convex case,
proving that $f$ is Gateaux differentiable amounts to observing that it is
semi-convex thanks to the Hopf-Lax representation and semi-concave thanks to
the fact it can be written as the limit of the enriched free energy (see [7,
Proposition 8.6]). Since the Hopf-Lax representation for viscosity solutions
is not available when $\xi$ is nonconvex, some innovative new ideas seem to be
needed to prove (or disprove) the Gateaux differentiability of $f$ in the
nonconvex case. Nevertheless, for nonconvex models, we can formulate the
following hypothesis.

(H${}_{\xi,P_{1}}$) $\begin{split}\text{The viscosity solution of \eqref{e.HJ
nonconvex} is Gateaux}\\\ \text{differentiable on
$(0,\infty)\times\left(\mathcal{Q}_{\uparrow}(\mathbb{R}^{D}_{+})\cap
L^{\infty}\right)$.}\end{split}$

In physics terms, (H${}_{\xi,P_{1}}$) can be regarded as the statement that no
first order phase transition occurs in the mean-field spin glass model with
covariance function $\xi$ and reference measure $P_{1}$. According to [7,
Propositions 7.2 & 8.6], when $\xi$ is convex, (H${}_{\xi,P_{1}}$) holds. When
$\xi$ is nonconvex, assuming (H${}_{\xi,P_{1}}$) holds, we obtain the
following representation for the limit free energy.

######  Theorem 7.10.

Assume that $\xi$ is permutation-invariant and only depends on the diagonal
coefficients of its argument, assume that $P_{1}$ is permutation-invariant. If
(H${}_{\xi,P_{1}}$) is true, then for every $t>0$, the free energy
$\overline{F}_{N}(t)$ converges as $N\to+\infty$ and

(7.12) $\lim_{N\to+\infty}\overline{F}_{N}(t)=g(t,0),$

where $g$ is the viscosity solution of (7.11). Furthermore, we have
$g(t,0)=f(t,0)$, where $f$ is the viscosity solution of (7.10) and

$g(t,0)=\sup_{p\in\mathcal{Q}(\mathbb{R}_{+})\cap
L^{\infty}}\left\\{\psi(p\text{id}_{D})-t\int_{0}^{1}\Xi^{*}\left(\frac{p(u)\text{id}_{D}}{t}\right)\mathrm{d}u\right\\}.$

######  Remark 7.11.

In fact, even if (H${}_{\xi,P_{1}}$) only holds on a region of the form
$[0,t_{c})\times(\mathcal{Q}_{\uparrow}(\mathbb{R}^{D}_{+})\cap L^{\infty})$
for some $t_{c}>0$ then (7.12) still holds for $t\leqslant t_{c}$.

######  Remark 7.12.

It seems plausible that there exists $t_{c}>0$ such that the weaker version of
(H${}_{\xi,P_{1}}$) mentioned in Remark 7.11 holds. Indeed, $\psi$ is Gateaux
differentiable on $\mathcal{Q}(\mathbb{R}^{D}_{+})\cap L^{2}$ and its gradient
is a Lipschitz function [7, Corollary 5.2]. Therefore, the characteristic
curves

$q\mapsto q-t\nabla\xi(\nabla\psi(q)),$

are injective when $t\geqslant 0$ is small enough
$(t<1/\|\nabla\xi(\nabla\psi)\|_{\text{Lip}})$. Usually, the viscosity
solution remains differentiable as long as the characteristic curves are
injective.

When $D=1$, the Parisi formula is a differentiable function of
$\beta=\sqrt{2t}$ [19]. Therefore, the right-hand side of (7.12) is
differentiable on $(0,+\infty)$ as a function of $t$. So at least
$\eqref{e.hypo}$ is not in direct contradiction with (7.12).

Conditionally on (H${}_{\xi,P_{1}}$), Theorem 7.10 confirms [16, Conjecture
2.6] for permutation-invariant models. Note that the argument given below will
not show that $(t,q)\mapsto\lim_{N\to+\infty}\overline{F}_{N}(t,q)$ is the
viscosity solution of

$\begin{cases}\partial_{t}f-\int\xi(\nabla f)=0\text{ on
}(0,\infty)\times(\mathcal{Q}(\mathbb{R}^{D}_{+})\cap L^{2})\\\
f(0,\cdot)=\psi\text{ on }\mathcal{Q}(\mathbb{R}^{D}_{+})\cap
L^{2}.\end{cases}$

In fact, even under (H${}_{\xi,P_{1}}$), it does not follow from the proof
below that $\overline{F}_{N}(t,q)$ converges as $N\to+\infty$ when $q$ is not
of the form $(p,\dots,p)$.

It is worth mentioning that Theorem 7.10 still holds if (H${}_{\xi,P_{1}}$) is
replaced by the assumption that the enriched free energy converges as
$N\to+\infty$ and the limit is Gateaux differentiable on
$(0,\infty)\times\left(\mathcal{Q}_{\uparrow}(\mathbb{R}^{D}_{+})\cap
L^{\infty}\right)$. To prove this version of Theorem 7.10, one can adapt the
argument given below, replacing $f$ the viscosity solution of (7.10) by the
limit free energy and appealing to [7, Proposition 7.2]. Note that when
choosing to state Theorem 7.10 this way, the convergence of the free energy as
$N\to+\infty$ is a hypothesis, not a conclusion.

###### Proof of Theorem 7.10.

Let $f$ and $g$ denote the viscosity solutions of (7.10) and (7.11). Given
(7.9), it suffices to verify that $f(t,0)=g(t,0)$ to prove the desired result.
As previously, for every $p\in\mathcal{Q}(\mathbb{R}_{+})\cap L^{1}$, we let

$f^{\dagger}(t,p)=f(t,(p,\dots,p)).$

Step 1. We show that for every $s\in\mathscr{S}_{D}$,

$f(t,(p_{1},\dots,p_{D}))=f(t,(p_{s(1)},\dots,p_{s(D)})).$

Given $s\in\mathscr{S}_{D}$, we let

$\displaystyle f^{s}(t,(p_{1},\dots,p_{D}))$
$\displaystyle=f(t,(p_{s(1)},\dots,p_{s(D)}))$
$\displaystyle\psi^{s}(p_{1},\dots,p_{D})$
$\displaystyle=\psi(p_{s(1)},\dots,p_{s(D)}).$

Since $\psi$ depends only on $P_{1}$ and not on $\xi$, according to
Proposition 5.1, we have $\psi^{s}=\psi$. Using the permutation invariance of
$\xi$, it can be checked that $f^{s}$ is a viscosity solution of (7.10). By
uniqueness of the viscosity solution of (7.10), we obtain $f^{s}=f$.

Step 2. We show that $f^{\dagger}$ is a strong solution of

$\begin{cases}\partial_{t}f^{\dagger}-\int\Xi_{\dagger}\left(\frac{\nabla
f^{\dagger}}{D}\right)=0\text{ on
}(0,+\infty)\times\left(\mathcal{Q}_{\uparrow}(\mathbb{R}_{+})\cap
L^{\infty}\right)\\\ f^{\dagger}(0,\cdot)=\psi^{\dagger}\text{ on
}\mathcal{Q}_{\uparrow}(\mathbb{R}_{+})\cap L^{\infty}.\end{cases}$

Using Step 1, we can proceed as in the proof of Proposition 5.7 to discover
that for every
$(t,p)\in(0,+\infty)\times(\mathcal{Q}_{\uparrow}(\mathbb{R}_{+})\cap
L^{\infty})$, $f^{\dagger}(t,\cdot)$ is Gateaux differentiable at $p$ and

(7.13) $\nabla f(t,(p,\dots,p))=\left(\frac{\nabla
f^{\dagger}(t,p)}{D},\dots,\frac{\nabla f^{\dagger}(t,p)}{D}\right).$

It then follows from (H${}_{\xi,P_{1}}$) that

$\partial_{t}f^{\dagger}(t,p)-\int\xi_{\dagger}\left(\frac{\nabla
f^{\dagger}(t,p)}{D}\right)=\partial_{t}f(t,(p,\dots,p))-\int\xi(\nabla
f(t,(p,\dots,p)))=0.$

Finally, since $\xi_{\dagger}=\Xi_{\dagger}$ we conclude Step 2.

Step 3. We show that $f(t,0)=g(t,0)$.

According to Proposition 7.2, $\Xi$ is convex on $\mathbb{R}^{D}_{+}$ and
$\Xi$ is the covariance function of the process defined by (7.7). In Theorem
5.9 we have shown that, when the nonlinearity is convex, any strong solution
of a Hamilton-Jacobi equation on $\mathcal{Q}_{\uparrow}(\mathbb{R}_{+})\cap
L^{\infty}$ is also a viscosity solution on $\mathcal{Q}(\mathbb{R}_{+})\cap
L^{2}$. Therefore, using Step 2 and applying Theorem 5.9 to $\Xi_{\dagger}$
and $f^{\dagger}$, we discover that $f^{\dagger}$ the viscosity solution of

$\begin{cases}\partial_{t}f^{\dagger}-\int\Xi_{\dagger}\left(\frac{\nabla
f^{\dagger}}{D}\right)=0\text{ on
}(0,+\infty)\times(\mathcal{Q}(\mathbb{R}_{+})\cap L^{2})\\\
f^{\dagger}(0,\cdot)=\psi^{\dagger}\text{ on }\mathcal{Q}(\mathbb{R}_{+})\cap
L^{2}.\end{cases}$

Once again, since $\Xi_{\dagger}=\xi_{\dagger}$, $f^{\dagger}$ is in fact the
viscosity solution of (7.11). Finally, by uniqueness of the viscosity
solution, it follows that $f^{\dagger}=g$ and choosing $p=0$, we deduce
$f(t,0)=g(t,0)$. Thus $F_{N}(t)$ converges as $N\to+\infty$ and

$\lim_{N\to+\infty}\overline{F}_{N}(t)=f(t,0)=g(t,0).$

Step 4. We show that

$g(t,0)=\sup_{p\in\mathcal{Q}(\mathbb{R}_{+})\cap
L^{\infty}}\left\\{\psi(p\text{id}_{D})-t\int_{0}^{1}\Xi^{*}\left(\frac{p(u)\text{id}_{D}}{t}\right)\mathrm{d}u\right\\}.$

According to Proposition 7.2, $\xi_{\dagger}$ is convex on $\mathbb{R}_{+}$.
Therefore, it follows from the one dimensional Hopf-Lax representation [9,
Proposition A.3] that,

$g(t,0)=\sup_{p\in\mathcal{Q}(\mathbb{R}_{+})\cap
L^{\infty}}\left\\{\psi^{\dagger}(p)-t\int_{0}^{1}\left(\xi_{\dagger}\left(\frac{\cdot}{D}\right)\right)^{*}\left(\frac{p(u)}{t}\right)\mathrm{d}u\right\\}.$

In addition, $\Xi$ is convex on $\mathbb{R}_{+}^{D}$ and we have

$\left(\xi_{\dagger}\left(\frac{\cdot}{D}\right)\right)^{*}(\lambda)=\Xi^{*}(\lambda,\dots,\lambda).$

Thus,

$g(t,0)=\sup_{p\in\mathcal{Q}(\mathbb{R}_{+})\cap
L^{\infty}}\left\\{\psi(p\text{id}_{D})-t\int_{0}^{1}\Xi^{*}\left(\frac{p(u)\text{id}_{D}}{t}\right)\mathrm{d}u\right\\}.$

∎

## References

  * [1] Antonio Auffinger and Wei-Kuo Chen.  The Parisi formula has a unique minimizer.  Comm. Math. Phys., 335(3):1429–1444, 2015.
  * [2] Adriano Barra, Giuseppe Genovese, Francesco Guerra, and Daniele Tantari.  How glassy are neural networks?  J. Stat. Mech., 2012(07):P07009, 2012.
  * [3] Erik Bates and Youngtak Sohn.  Free energy in multi-species mixed p-spin spherical models.  Electron. J. Probab., 27:1–75, 2022.
  * [4] Erik Bates and Youngtak Sohn.  Parisi formula for balanced Potts spin glass.  Preprint, arXiv:2310.06745.
  * [5] Hong-Bin Chen.  On Parisi measures of Potts spin glasses with correction.  Preprint, arXiv:2311.11699.
  * [6] Hong-Bin Chen.  Parisi PDE and convexity for vector spins.  Preprint, arXiv:2311.10446.
  * [7] Hong-Bin Chen and Jean-Christophe Mourrat.  On the free energy of vector spin glasses with non-convex interactions.  Preprint, arXiv:2311.08980.
  * [8] Hong-Bin Chen and Jiaming Xia.  Fenchel-Moreau identities on convex cones.  Preprint, arXiv:2011.06979.
  * [9] Hong-Bin Chen and Jiaming Xia.  Hamilton-Jacobi equations from mean-field spin glasses.  Preprint, arXiv:2201.12732.
  * [10] Hong-Bin Chen and Jiaming Xia.  Hamilton-Jacobi equations with monotone nonlinearities on convex cones.  Preprint, arXiv:2206.12537.
  * [11] David Elderfield and David Sherrington.  The curious case of the Potts spin glass.  J. Phys. C: Solid State Phys., 16(15):497–503, 1983.
  * [12] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, second edition, 2010.
  * [13] Francesco Guerra.  Broken replica symmetry bounds in the mean-field spin glass model.  Comm. Math. Phys., 233(1):1–12, 2003.
  * [14] Francesco Guerra and Fabio Lucio Toninelli.  The thermodynamic limit in mean-field spin glass models.  Comm. Math. Phys., 230:71–79, 2002.
  * [15] Jean-Christophe Mourrat.  Nonconvex interactions in mean-field spin glasses.  Probab. Math. Phys., 2(2):281–339, 2021.
  * [16] Jean-Christophe Mourrat.  The Parisi formula is a Hamilton-Jacobi equation in Wasserstein space.  Canad. J. Math., 74(3):607–629, 2022.
  * [17] Jean-Christophe Mourrat.  Free energy upper bound for mean-field vector spin glasses.  Ann. Inst. Henri Poincaré Probab. Stat., 59(3):1143–1182, 2023.
  * [18] Ben O’Neill.  The double-constant matrix, centering matrix and equicorrelation matrix: Theory and applications.  Preprint, arXiv:2109.05814.
  * [19] Dmitry Panchenko.  On differentiability of the Parisi formula.  Elect. Comm. in Probab., 13:241–247, 2008.
  * [20] Dmitry Panchenko.  The Sherrington-Kirkpatrick model.  Springer Monographs in Mathematics. Springer, New York, 2013.
  * [21] Dmitry Panchenko.  The free energy in a multi-species Sherrington-Kirkpatrick model.  Ann. Probab., 43(6):3494–3513, 2015.
  * [22] Dmitry Panchenko.  Free energy in the Potts spin glass.  Ann. Probab., 46(2):829–864, 2018.
  * [23] Dmitry Panchenko.  Free energy in the mixed $p$-spin models with vector spins.  Ann. Probab., 46(2):865–896, 2018.
  * [24] Giorgio Parisi.  Infinite number of order parameters for spin-glasses.  Phys. Rev. Lett., 43(23):1754–1756, 1979.
  * [25] Ralph Tyrrell Rockafellar.  Convex analysis, volume 36.  Princeton university press, 1970.
  * [26] Michel Talagrand.  The Parisi formula.  Ann. of Math., 163(1):221–263, 2006.