aainstitutetext: Dipartimento di Matematica e Fisica Ennio De Giorgi,
Università del Salento, Via per Arnesano, 73100, Lecce, Italybbinstitutetext:
Scuola Superiore ISUFI, Campus Ecotekne, Via Monteroni, 73100,
Lecce,Italyccinstitutetext: Istituto Nazionale di Fisica Nucleare, Campus
Ecotekne, Via Monteroni, 73100, Lecce,Italy

# Rigorous approaches for spin glass and Gaussian spin glass with P-wise
interactions

Linda Albanese a, b Andrea Alessandrelli

###### Abstract

Purpose of this paper is to face up to P-spin glass and Gaussian P-spin model,
i.e. spin glasses with polynomial interactions of degree $P>2$. We consider
the replica symmetry and first step of replica simmetry breaking assumptions
and we solve the models via transport equation and Guerra’s interpolating
technique, showing that we reach the same results.

Thus, using rigorous approaches, we recover the same expression for quenched
statistical pressure and self-consistency equation in both assumption found
with other techniques, including the well-known replica trick technique.

At the end, we show that for $P=2$ the Gaussian P-spin glass model is
intrinsecally RS.

## Introduction

In last few decades statistical mechanics has played an important role in
different fields, e.g. spin glassesMPV ; Bovier1  and neural networksAmit ;
PRLNN ; leonelli . For this reason there has been a lot of research to tackle
increasingly complex models through rigorous mathematical techniqueBovier2 ,
alternative to the well-known replica-trickMPV . Furthermore the growing
interest in Replica Symmetry Breaking and its connection with ultrametricity,
as just proved in PanchenkoUltra ; GG , has taken the Scientific Community to
take into account the results linked to this assumptionTala_break ; Zecchina1
; Remi1 ; Zecchina2 .

In this paper we focus on mathematical methods on spin glasses, considered a
real challenge for mathematicianstala . In particular, in this paper we want
to generalize some computations for Sherrington Kirkpatrick and Gaussian
models with two-spin interactions in a polynomial interactions of degree
$P>2$, with two different rigorous mathematical approaches, namely Guerra’s
interpolating technique, introduced in GuerraSum ; Guerra , and transport
equation, introduced in Replica Symmetric assumption in AABF-NN2020  and in
Replica Symmetry Breaking lindaRSB  for the description of Hopfield model and,
afterwards, for deep Hopfield networks trans_DHN . Thus we prove that both
methods are consistent and mathematically valid for these models.

Furthermore we face to RS and 1RSB assumptions, showing that, as fas as
Gaussian spin glass model concernes, we reach the results in AdrianoGauss ;
glassy  for $P=2$ and in crisanti ; gardner ; Tala3  for $P>2$. For the 1RSB
assumption, we use the broken replica interpolation introduced by Francesco
Guerra for the Sherrington Kirkpatrick model Guerra  and, moreover, we adapt
it into a PDE framework (standard transport equation).

We stress that the present paper is focused on mathematical methods to deepen
these models. So, we do not face up to physical interpretation of our results,
for which we remind to crisanti ; GrossMezard ; Tala3 .

The paper is structured as follows.

In the first section we analyze P-spin glass model, namely Sherrington
Kirkpatrick with P-wise interactions, starting from the main characters until
the expression in RS and 1RSB assumption of quenched statistical pressure and
self-consistency equations. For both assumption we compute via Guerra’s
interpolating technique and transport equation, showing that we reach the same
results. We proceed in similar way for Gaussian P-spin glass in the second
section, coming to the emergent property for this model too. The paper is
closed by a section concerning the case $P=2$ for Gaussian P-spin glass, in
which we show that it is intrinsecally RS, and some appendices with
computations from results of the paper.

##  1 P-spin glass model

In this section we deal with P-spin glass model, namely Sherrington-
Kirkpatrick (SK) model with $P$-wise interactions, introduced by Derrida in
Derrida  and meticulously described afterwards, e.g. gardner ; Tala4 ; Tala5 ;
GrossMezard ; MPV . In particular we find in RS and 1RSB assumption the
expressions of statistical quenched pressure and self-consistency equations.

###  1.1 Generalities

######  Definition 1.

Let $P\in\mathbb{N}$ and $\bm{\sigma}\in\\{-1;+1\\}^{N}$ be a configuration of
N spins, the Hamiltonian of the P-spin glass model is defined as

$H_{N}^{(P)}(\bm{\sigma}|\bm{J})\coloneqq-\dfrac{1}{P!}\displaystyle\sum\limits_{i_{1},\cdots,i_{P}=1}^{N,\cdots,N}\,J_{i_{1},\cdots,i_{P}}\sigma_{i_{1}}\cdots\sigma_{i_{P}}$
(1)

where the P-wise quenched couplings
$\bm{J}=\\{J_{i_{1},\cdots,i_{P}}\\}_{i_{1},\cdots,i_{P}=1,\cdots,N}$ are
given by

$J_{i_{1},...,i_{P}}\coloneqq\,\dfrac{J_{0}}{N^{P-1}}+J\sqrt{\dfrac{2}{N^{P-1}}}z_{i_{1},\cdots,i_{P}}$
(2)

with $J_{0}\in\mathbb{R}^{+}$ and $z_{i_{1},\cdots,i_{P}}$ i.i.d. standard
random variables $z_{i_{1},\cdots,i_{P}}\sim N(0;1)$.

######  Definition 2.

The partition function related to the Hamiltonian (1) is given by

$\begin{array}[]{lll}Z_{N}^{(P)}(\beta,\bm{J})&\coloneqq\sum_{\bm{\sigma}}\exp\left[-\beta
H_{N}^{(P)}(\bm{\sigma}|\bm{J})\right]\,,\end{array}$ (3)

where $\beta\in\mathbb{R}^{+}$ is the inverse temperature in proper units such
that for $\beta\to 0$ the probability distribution for the neural
configuration is uniformly spread while for $\beta\to\infty$ it is sharply
peaked at the minima of the energy function (1).

We introduce the _Boltzmann average_ induced by the partition function (3),
denoted with $\omega_{\bm{J}}$ and, for an arbitrary observable
$O(\bm{\sigma})$, defined as

$\omega_{\bm{J}}(O(\bm{\sigma})):=\frac{\sum_{\bm{\sigma}}O(\bm{\sigma})e^{-\beta
H_{N}(\bm{\sigma}|\bm{J})}}{Z_{N}(\beta,\bm{J})}.$ (4)

This can be further averaged over the realization of the
$J_{i_{1},\cdots,i_{P}}$’s (also referred to as _quenched average_) to get

$\langle
O(\bm{\sigma})\rangle\coloneqq\mathbb{E}\omega_{\bm{J}}(O(\bm{\sigma})).$ (5)

######  Definition 3.

The intensive quenched pressure of the P-spin glass model (1) is defined as

$\mathcal{A}_{N}^{(P)}(\beta,\bm{J})\coloneqq\frac{1}{N}\mathbb{E}\ln
Z_{N}^{(P)}(\beta,\bm{J}),$ (6)

and its thermodynamic limit, assuming its existence, is referred to as

$\mathcal{A}^{(P)}(\beta,\bm{J})\coloneqq\lim_{N\to\infty}\mathcal{A}^{(P)}_{N}(\beta,\bm{J}).$
(7)

######  Remark 1.

The existence of the thermodynamical limit of free energy density is proved
for P-spin glass model in Guerra and Toninelli’s workGuerraTon .

######  Remark 2.

We stress that the quenched statistical pressure $\mathcal{A}^{(P)}$ is
equivalent to free energy, namely

$\displaystyle\mathcal{A}^{(P)}(\beta,\bm{J})=-{\beta}f^{(P)}(\beta,\bm{J})=-{\beta}\left(E(\beta,\bm{J})-\frac{S(\beta,\bm{J})}{\beta}\right)$
(8)

where $E$ is the internal energy and S the entropy of the system.

In order to solve the model we want to find out an explicit expression for the
quenched pressure (7) in terms of the natural order parameters of the theory,
namely the magnetization $m$ and the two-replica overlap $q_{12}$, defined in
the following

######  Definition 4.

The order parameters used to describe the macroscopic behavior of the model
are the standard ones Amit ; Coolen , namely, the magnetization

$m\coloneqq\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}\,,$ (9)

and the two-replica overlap, introduced as

$q_{12}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{(1)}\sigma_{i}^{(2)}\,.$
(10)

###  1.2 Resolution via Guerra’s interpolation

The purpose of this section is to solve the P-spin glass model through
Guerra’s interpolating technique. To do so, we compute the derivative w.r.t.
the interpolating parameter $t$ and we apply the Fundamental theorem of
Calculus. In the end, we find the expression of statistical pressure in the
approximation of replica symmetry (RS) and first step of replica symmetry
breaking (RSB).

####  1.2.1 RS solution

######  Definition 5.

Under the replica-symmetry assumption, the order parameters, in the
thermodynamic limit, self-average around their mean values and their
distributions get delta-peaked at their equilibrium value (denoted with a
bar), independently of the replica considered, namely

$\displaystyle\lim_{N\to\infty}\langle(m-\bar{m})^{2}\rangle=0$
$\displaystyle\Rightarrow$ $\displaystyle\lim_{N\to\infty}\langle
m\rangle=\bar{m},$ (11)
$\displaystyle\lim_{N\to\infty}\langle(q_{12}-\bar{q})^{2}\rangle=0$
$\displaystyle\Rightarrow$ $\displaystyle\lim_{N\to\infty}\langle
q_{12}\rangle=\bar{q}.$ (12)

For the generic order parameter $X$ this can be rewritten as $\langle(\Delta
X)^{2}\rangle\xrightarrow[]{N\to\infty}0$, where

$\Delta X\coloneqq X-\bar{X},$

and, clearly, the RS approximation also implies that, in the thermodynamic
limit, $\langle\Delta X\Delta Y\rangle=0$ for any generic pair of order
parameters $X,Y$. Moreover in the thermodynamic limit, we have $\langle(\Delta
X)^{k}\rangle\rightarrow 0$ for $k\geq 2$.

######  Definition 6.

Given the interpolating parameter $t\in[0,1]$, $A,\ \psi\in\mathbb{R}$ and
$z_{i}\sim\mathcal{N}(0,1)$ for $i=1,\ldots,N$ standard Gaussian variables
i.i.d., the Guerra’s interpolating partition function is given as

$\begin{array}[]{lll}\mathcal{Z}^{(P)}_{N}(t)&\coloneqq&\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\exp{}\Bigg{[}t\dfrac{\beta^{{}^{\prime}}J_{0}N}{2}m^{P}(\bm{\sigma})+(1-t)N\psi\,m(\bm{\sigma})+\\\
\\\
&&+\sqrt{t}\beta^{{}^{\prime}}J\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{i_{1}\cdots,i_{{}_{P}}}\sigma_{i_{1}}\cdots\sigma_{{}_{P}}+\\\
\\\
&&+A\sqrt{1-t}\,\displaystyle\sum\limits_{i=1}^{N}z_{i}\sigma_{i}\,,\end{array}$
(13)

where $\beta^{{}^{\prime}}=2\beta/P!$.

######  Definition 7.

The interpolating pressure for the P-spin glass model (1), at finite $N$, is
introduced as

$\displaystyle\mathcal{A}^{(P)}_{N}(t)$ $\displaystyle\coloneqq$
$\displaystyle\frac{1}{N}\mathbb{E}\left[\ln\mathcal{Z}^{(P)}_{N}(t)\right],$
(14)

where the expectation $\mathbb{E}$ is now meant over $z_{i_{1},\cdots,i_{P}}$
and $z_{i}$, in the thermodynamic limit,

$\mathcal{A}^{(P)}(t)\coloneqq\lim_{N\to\infty}\mathcal{A}^{(P)}_{N}(t).$ (15)

By setting $t=1$ the interpolating pressure recovers the original one (6),
that is $A_{N}^{(P)}(\beta,J_{0},J)=\mathcal{A}^{(P)}_{N}(t=1)$.

######  Remark 3.

The interpolating structure implies an interpolating measure, whose related
Boltzmann factor reads as

$\mathcal{B}(\bm{\sigma};t)\coloneqq\exp\left[\beta\mathcal{H}(\bm{\sigma};t)\right];$
(16)

In this way the partition function is written as
$\mathcal{Z}_{N}(t)=\sum_{\bm{\sigma}}\mathcal{B}(\bm{\sigma};t)$ .
A generalized average follows from this generalized measure as

$\omega_{t}(O(\bm{\sigma}))\coloneqq\sum_{\bm{\sigma}}O(\bm{\sigma})\mathcal{B}(\bm{\sigma};t)$
(17)

and

$\langle
O(\bm{\sigma})\rangle_{t}\coloneqq\mathbb{E}[\omega_{t}(O(\bm{\sigma}))].$
(18)

where $\mathbb{E}$ denotes the average over $z_{i_{1},\cdots,i_{P}}$ and
$\\{z_{i}\\}_{i=1,\cdots,N}$.

Of course, when $t=1$ the standard Boltzmann measure and related average is
recovered. Hereafter, in order to lighten the notation, we will drop the
subindices $t$.

######  Lemma 1.

The $t$ derivative of interpolating pressure is given by

$\small\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}\coloneqq&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\left(\langle
m^{P}\rangle-\dfrac{2\psi}{\beta^{{}^{\prime}}J_{0}}\langle
m\rangle\right)-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(\langle
q_{12}^{P}\rangle-\dfrac{2A^{2}}{\beta^{\prime}\,{}^{2}J^{2}}\langle
q_{12}\rangle\right)\\\ \\\
&+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}-\dfrac{A^{2}}{2}\end{array}$ (19)

###### Proof.

Deriving equation (14) with respect to $t$, we get

$\footnotesize\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}&=&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle-\psi\langle m\rangle+\\\ \\\
&&+\dfrac{\beta^{{}^{\prime}}J}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}z_{\bm{i}}\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})\right]+\\\
\\\
&&-\dfrac{1}{2N\sqrt{1-t}}\mathbb{E}\left[A\sum\limits_{i}z_{i}\omega(\sigma_{i})\right]\,.\end{array}$
(20)

Now, using the Stein’s lemma (also known as Wick’s theorem) on standard
Gaussian variable $z_{i}$ and $z_{\bm{i}}$, which is for a standard Gaussian
variable $J$, i.e. $J\sim N(0,1)$, and for a generic function $f(J)$ for which
$\mathbb{E}zf(z)$ and $\mathbb{E}\partial_{J}f(J)$ both exist, then

$\displaystyle\mathbb{E}\left(Jf(J)\right)=\mathbb{E}\left(\frac{\partial
f(J)}{\partial J}\right)\,,$ (21)

we may rewrite the second and the third member of (20) as

$\footnotesize\begin{array}[]{lll}\dfrac{\beta^{{}^{\prime}}J}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}z_{\bm{i}}\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})\right]-\dfrac{1}{2N\sqrt{1-t}}\mathbb{E}\left[A\sum\limits_{i}z_{i}\omega(\sigma_{i})\right]=D_{1}+D_{2}\,.\end{array}$

Let’s investigate those three terms:

$\footnotesize\begin{array}[]{lll}D_{1}&=\dfrac{\beta^{{}^{\prime}}J}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}\partial_{z_{\bm{i}}}\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})\right]\\\
\\\
&=\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4N^{P}}\left(\sum\limits_{\bm{i}}\mathbb{E}\left[\omega((\sigma_{i_{1}}\cdots\sigma_{i_{P}})^{2})\right]-\sum\limits_{\bm{i}}\mathbb{E}\left[\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})^{2}\right]\right)\\\
\\\ &=\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\Big{[}1-\langle
q_{12}^{P}\rangle\Big{]}\,;\end{array}$ (22)
$\footnotesize\begin{array}[]{lll}D_{2}&=-\dfrac{1}{2N\sqrt{1-t}}A\sum\limits_{i}\mathbb{E}\left[\partial_{z_{i}}\omega(\sigma_{i})\right]\\\
\\\
&=-\dfrac{1}{2N}A^{2}\left(\sum\limits_{i}\mathbb{E}\left[\omega(\sigma^{2}_{i})\right]-\sum\limits_{i}\mathbb{E}\left[\omega(\sigma_{i})^{2}\right]\right)\\\
\\\ &=-\dfrac{1}{2}A^{2}\Big{[}1-\langle q_{12}\rangle\Big{]}\,.\end{array}$
(23)

Rearranging together (22) and (23) we obtain the thesis. ∎

######  Remark 4.

We stress that, for the RS assumption presented in Definition (5), we can use
the relations

$\displaystyle\langle m^{P}\rangle$
$\displaystyle=\sum_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(m-\bar{m})^{k}\rangle\bar{m}^{P-k}+\bar{m}^{P}(1-P)+P\bar{m}^{P-1}\langle
m\rangle\,,$ (24) $\displaystyle\langle q_{12}^{P}\rangle$
$\displaystyle=\sum_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(q_{12}-\bar{q})^{k}\rangle\bar{q}^{P-k}+\bar{q}^{P}(1-P)+P\bar{q}^{P-1}\langle
q_{12}\rangle\,,$ (25)

which are proven in Appendix (A). Using these relations, if we fix the
constants as

$\begin{array}[]{lll}\psi=\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{{}^{P-1}}\,,\\\
\\\
A^{2}=\dfrac{P}{2}\beta^{\prime}\,{}^{2}J^{2}\bar{q}^{{}^{P-1}}\,,\end{array}$
(26)

the (19) in the thermodynamical limit reads as

$\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}&\coloneqq&-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right)\end{array}$
(27)

which is now independent of $t$.

######  Proposition 1.

In the thermodynamic limit ($N\to\infty$) and under RS assumption, applying
the Fundamental Theorem of Calculus and using the suitable values of $A$ and
$\psi$, we find the quenched pressure for the P-spin glass model as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(\beta,J_{0},J)&=&\ln{2}+\left\langle\ln{\cosh{\left[\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{{}^{P-1}}+z\beta^{{}^{\prime}}J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}}\right\rangle_{z}+\\\
\\\
&&-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right)\,,\end{array}$
(28)

where the values of order parameters are ruled by the following self-
consistence equations

$\begin{array}[]{lll}\bar{m}=\left\langle\tanh{\left[\dfrac{P}{2}\beta
J_{0}\bar{m}^{{}^{P-1}}+z\beta
J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}\right\rangle_{z}\,,\\\ \\\
\bar{q}=\left\langle\tanh{}^{2}{\left[\dfrac{P}{2}\beta
J_{0}\bar{m}^{{}^{P-1}}+z\beta
J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}\right\rangle_{z}\,.\end{array}$
(29)

###### Proof.

Using the Fundamental Theorem of Calculus:

$\footnotesize\mathcal{A}^{(P)}=\mathcal{A}^{(P)}(t=1)=\mathcal{A}^{(P)}(t=0)+\int\limits_{0}^{1}\,\partial_{s}\mathcal{A}^{(P)}(s)\Big{|}_{s=t}\,dt$
(30)

and computing the one-body terms

$\footnotesize\begin{array}[]{lll}\mathcal{A}^{(P)}(t=0)&=&\dfrac{1}{N}\mathbb{E}\ln{}\Bigg{\\{}\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\exp{}\Bigg{[}N\psi\,m(\bm{\sigma})+A\,\displaystyle\sum\limits_{i=1}^{N}z_{i}\sigma_{i}\Bigg{]}\Bigg{\\}}\,,\\\
\\\
&=&\ln{2}+\left\langle\ln{\cosh{\left[\psi+zA\right]}}\right\rangle_{z}+\\\
\\\
&=&\ln{2}+\left\langle\ln{\cosh{\left[\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{{}^{P-1}}+z\beta^{{}^{\prime}}J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}}\right\rangle_{z}\,.\end{array}$
(31)

Finally, putting (27) and (31) in (30), we find (28).

Extremizing the statistical pressure in (28) w.r.t. the order parameters we
find the self-consistency equations. ∎

######  Remark 5.

If we do not take into account the disordered part of the system, namely
$J=0$, in (28), we recover for this P-spin ferromagnetic model the same result
in fachechi .

######  Remark 6.

If we don’t consider in (28) the ferromagnetic part of the system, namely
$J_{0}=0$, we recover the same expressions found in Adriano_HJ .

####  1.2.2 1RSB solution

In this subsection we turn to the solution of the P-spin glass model via the
Guerra’s interpolating technique, restricting the description at the first
step of RSB.

######  Definition 8.

In the first step of replica-symmetry breaking, the distribution of the two-
replica overlap $q$, in the thermodynamic limit, displays two delta-peaks at
the equilibrium values, referred to as $\bar{q}_{1},\ \bar{q}_{2}$, and the
concentration on the two values is ruled by $\theta\in[0,1]$, namely

$\lim_{N\rightarrow+\infty}P^{\prime}_{N}(q)=\theta\delta(q-\bar{q}_{1})+(1-\theta)\delta(q-\bar{q}_{2}).$
(32)

The magnetization still self-averages at $\bar{m}$ as in (11).

Following the same route pursued in the previous sections, we need an
interpolating partition function $\mathcal{Z}$ and an interpolating quenched
pressure $\mathcal{A}$, that are defined hereafter.

######  Definition 9.

Given the interpolating parameter $t$ and the i.i.d. auxiliary fields
$\\{z_{i}^{(1)},z_{i}^{(2)}\\}_{i=1,...,N}$, with
$z_{i}^{(1,2)}\sim\mathcal{N}(0,1)$ for $i=1,...,N$ we can write the 1-RSB
interpolating partition function $\mathcal{Z}_{N}(t)$ for the P-spin glass
model (1) recursively, starting by

$\begin{array}[]{lll}\mathcal{Z}^{(P)}_{2}(t)&:=&\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\displaystyle\int\mathcal{D}\bm{\tau}\exp{}\Bigg{[}t\dfrac{\beta^{{}^{\prime}}J_{0}N}{2}m^{P}(\bm{\sigma})+(1-t)N\psi\,m(\bm{\sigma})+\\\
\\\
&&+\sqrt{t}\beta^{{}^{\prime}}J\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{i_{1}\cdots,i_{{}_{P}}}\sigma_{i_{1}}\cdots\sigma_{i_{P}}+\\\
\\\
&&+\sqrt{1-t}\displaystyle\sum\limits_{a=1}^{2}\left(A_{a}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{(a)}\sigma_{i}\right)\end{array}$
(33)

where the $z_{i_{1}\cdots,i_{{}_{P}}}$’s i.i.d. standard Gaussian. Averaging
out the fields recursively, we define

$\displaystyle\mathcal{Z}_{1}^{(P)}(t)\coloneqq$
$\displaystyle\mathbb{E}_{2}\left[\mathcal{Z}_{2}^{(P)}(t)^{\theta}\right]^{1/\theta}$
(34) $\displaystyle\mathcal{Z}_{0}^{(P)}(t)\coloneqq$
$\displaystyle\exp\mathbb{E}_{1}\left[\ln\mathcal{Z}_{1}^{(P)}(t)\right]$ (35)
$\displaystyle\mathcal{Z}_{N}^{(P)}(t)\coloneqq$
$\displaystyle\mathcal{Z}_{0}^{(P)}(t),$ (36)

where with $\mathbb{E}_{a}$ we mean the average over the variables
$z_{i}^{(a)}$’s, for $a=1,2$, and with $\mathbb{E}_{0}$ we shall denote the
average over the variables $z_{i_{1}\cdots,i_{{}_{P}}}$’s.

######  Definition 10.

The 1RSB interpolating pressure, at finite volume $N$, is introduced as

$\mathcal{A}_{N}^{(P)}(t)\coloneqq\frac{1}{N}\mathbb{E}_{0}\big{[}\ln\mathcal{Z}_{0}^{(P)}(t)\big{]},$
(37)

and, in the thermodynamic limit, assuming its existence

$\mathcal{A}^{(P)}(t)\coloneqq\lim_{N\to\infty}\mathcal{A}^{(P)}_{N}(t).$ (38)

By setting $t=1$, the interpolating pressure recovers the standard pressure
(6), that is, $A^{(P)}_{N}(\beta,J_{0},J)=\mathcal{A}^{(P)}_{N}(t=1)$.

######  Remark 7.

In order to lighten the notation, hereafter we use the following

$\displaystyle\langle m\rangle=$
$\displaystyle\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\left[\mathcal{W}_{2}\frac{1}{N}\sum_{i=1}^{N}\omega(\sigma_{i})\right]$
(39) $\displaystyle\langle q_{12}\rangle_{1}=$
$\displaystyle\mathbb{E}_{0}\mathbb{E}_{1}\left[\frac{1}{N}\sum_{i=1}^{N}\left(\mathbb{E}_{2}\left[\mathcal{W}_{2}\omega(\sigma_{i})\right]\right)^{2}\right]$
(40) $\displaystyle\langle q_{12}\rangle_{2}=$
$\displaystyle\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\left[\mathcal{W}_{2}\frac{1}{N}\sum_{i=1}^{N}\omega(\sigma_{i})^{2}\right]$
(41)

where the weight $\mathcal{W}_{2}$ is defined as

$\mathcal{W}_{2}:=\frac{\mathcal{Z}_{2}^{\theta}}{\mathbb{E}_{2}\left[\mathcal{Z}_{2}^{\theta}\right]}.$
(42)

Now the next step is computing the $t$-derivative of the statistical pressure.
In this way we can apply the fundamental theorem of calculus and find the
solution of the original model.

######  Lemma 2.

The derivative w.r.t. $t$ of interpolating statistical pressure can be written
as

$\small\begin{array}[]{lll}d_{t}\mathcal{A}^{(P)}_{N}=&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle-\psi\langle
m\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1+(\theta-1)\langle
q_{12}^{P}\rangle_{2}-\theta\langle q_{12}^{P}\rangle_{1}\right]\\\ \\\
&-(A_{1})^{2}\left[1+(\theta-1)\langle q_{12}\rangle_{2}-\theta\langle
q_{12}\rangle_{1}\right]-(A_{2})^{2}\left[1+(\theta-1)\langle
q_{12}\rangle_{2}\right]\end{array}$ (43)

Since the proof is a bit lenghty, we leave it in Appendix (B).

######  Remark 8.

If we want to use the 1RSB assumpion in such a way that for $a=1,2$

$\displaystyle\langle m^{P}\rangle$
$\displaystyle=\sum_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(m-\bar{m})^{k}\rangle\bar{m}^{P-k}+\bar{m}^{P}(1-P)+P\bar{m}^{P-1}\langle
m\rangle\,,$ (44) $\displaystyle\langle q_{12}^{P}\rangle_{a}$
$\displaystyle=\sum_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(q_{12}-\bar{q}_{a})^{k}\rangle_{a}\bar{q}_{a}^{P-k}+\bar{q}_{a}^{P}(1-P)+P\bar{q}_{a}^{P-1}\langle
q_{12}\rangle_{a}\,,$ (45)

we need to fix the constants as

$\displaystyle\psi$
$\displaystyle=\frac{\beta^{{}^{\prime}}J_{0}}{2}P\bar{m}^{P-1},$ (46)
$\displaystyle A_{1}$
$\displaystyle=\frac{\beta^{\prime}\,{}^{2}J^{2}P}{4}\bar{q}_{1}^{P-1},$ (47)
$\displaystyle A_{2}$
$\displaystyle=\frac{\beta^{\prime}\,{}^{2}J^{2}P}{4}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}).$
(48)

Thus, the derivative w.r.t. $t$ in the thermodynamic limit is computed as

$\begin{array}[]{lll}d_{t}\mathcal{A}^{(P)}_{N}=&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P}(1-P)+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}\right]\\\
\\\
&-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}(P-1)\theta(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\,.\end{array}$
(49)

######  Proposition 2.

At finite size and under 1RSB assumption applying the Fundamental Theorem of
Calculus and using the suitable values of $A^{(1)},A^{(2)},\psi$, we find the
quenched pressure for the P-spin glass model as

$\begin{array}[]{lll}&\mathcal{A}^{(P)}=\log
2+\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P}(1-P)+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}\right]\\\
\\\
&+\dfrac{1}{\theta}\mathbb{E}_{1}\log\mathbb{E}_{2}\cosh^{\theta}\left[\beta^{{}^{\prime}}\left(\dfrac{J_{0}}{2}P\bar{m}^{P-1}+z^{(1)}J\sqrt{\frac{P}{2}\bar{q}_{1}^{P-1}}+z^{(2)}J\sqrt{\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}})\right)\right]\\\
\\\
&-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}(P-1)\theta(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})+\dfrac{\beta^{{}^{\prime}}J_{0}P}{2}\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(\Delta m)^{k}\rangle\bar{m}^{P-k}\\\ \\\
&+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}(\theta-1)\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(\Delta
q_{2})^{k}\rangle_{2}\bar{q}_{2}^{P-k}-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\theta\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(\Delta
q_{1})^{k}\rangle_{1}\bar{q}_{1}^{P-k}\end{array}$ (50)

where we use $\Delta X_{a}=X-\bar{X}_{a}$.

###### Proof.

We use the fundamental theorem of Calculus

$\displaystyle\mathcal{A}^{(P)}=\mathcal{A}^{(P)}(t=1)=\mathcal{A}^{(P)}(t=0)+\int_{0}^{1}\partial_{s}\mathcal{A}^{(P)}(s)|_{s=t}dt.$
(51)

The last step we need is the computation of one-body term. We omit it since it
is similar to RS case. ∎

######  Theorem 1.

In the thermodynamic limit, under 1RSB assumption, the expression of quenched
statistical pressure is

$\begin{array}[]{lll}\mathcal{A}^{(P)}=&\log
2-\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P}(P-1)+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}\right]\\\
\\\
&-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}(P-1)\theta(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})+\dfrac{1}{\theta}\mathbb{E}_{1}\log\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\,,\end{array}$
(52)

where

$\displaystyle\small g(\bm{z},\bar{m})$
$\displaystyle=\frac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P-1}+z^{(1)}\beta^{{}^{\prime}}J\sqrt{\frac{P}{2}\bar{q}_{1}^{P-1}}+z^{(2)}\beta^{{}^{\prime}}J\sqrt{\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}})$
(53)

where the order parameters are ruled by

$\displaystyle\bar{m}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh
g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]$ (54)
$\displaystyle\bar{q}_{1}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh
g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]^{2}$
(55)
$\displaystyle\bar{q}_{2}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh^{2}g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]$
(56)

where $g(\bm{z},\bar{m})$ is the same defined in (53).

###### Proof.

If we use definition 5, the last three terms of expression
(LABEL:A_1RSB_finite) will vanish in the thermodynamic limit, so we recover
(52).

Extremizing the statistical pressure in (52) w.r.t. the order parameters we
find the self-consistency equations. ∎

######  Remark 9.

The expression (52) with $J_{0}=0$ is the same computed by gardner ;
GrossMezard  through replica trick and by Adriano_HJ  via Hamilton-Jacobi
technique in 1RSB assumption.

###  1.3 Resolution via transport equation

Now the aim is to show that, using a different strategy, such as the transport
equation technique, already introduced in lindaRSB  for both assumption in
classic SK model with ferromagnetic contribution, we are able to retrieve the
same expression of quenched statistical pressure and self-consistency
equations found via Guerra’s interpolating technique.

####  1.3.1 RS solution

In this subsection we find the RS solution of P-spin glass model via the
transport equation technique. The strategy is to introduce an interpolating
pressure $\mathcal{A}_{N}$ living in a fictitious space-time framework and
recovering the intensive quenched pressure $\mathcal{A}_{N}$ of the original
model in a specific point of this space, and to show that it fulfills a
transport equation in such a way that the solution of the statistical
mechanical problem is recast in the solution of a partial differential
equation.

The definition of RS assumption for the order parameters is the same as (5).

######  Definition 11.

Given the interpolating parameter $t,x,w$, and $z_{i}\sim\mathcal{N}(0,1)$
standard Gaussian variables iid, the partition function is given as

$\footnotesize\begin{array}[]{lll}\mathcal{Z}^{(P)}_{N}(t,\bm{r})&\coloneqq&\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\exp{}\Bigg{[}t\dfrac{\beta^{{}^{\prime}}J_{0}N}{2}m^{P}(\bm{\sigma})+wN\,m(\bm{\sigma})+\\\
\\\
&&+\sqrt{t}\beta^{{}^{\prime}}J\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{i_{1}\cdots,i_{{}_{P}}}\sigma_{i_{1}}\cdots\sigma_{{}_{P}}+\\\
\\\
&&+\sqrt{x}\,\displaystyle\sum\limits_{i=1}^{N}z_{i}\sigma_{i}\,,\end{array}$
(57)

where we set $\beta^{\prime}=2\beta/P!$.

Similar to RS Guerra’s interpolation, we can define the interpolating
pressure, the Boltzmann factor and the generalized measure.

######  Lemma 3.

The partial derivatives of the interpolating pressure (14) w.r.t. $t,x,w$ give
the following expectation values:

$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial t}$
$\displaystyle=$ $\displaystyle\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\Big{(}1-\langle
q_{12}^{P}\rangle\Big{)},$ (58)
$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial x}$
$\displaystyle=$ $\displaystyle\dfrac{1}{2}\Big{(}1-\langle
q_{12}\rangle\Big{)},$ (59)
$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial w}$
$\displaystyle=$ $\displaystyle\langle m\rangle.$ (60)

###### Proof.

Since the procedures for the derivatives w.r.t. each parameter are analogous,
we prove only the derivatives w.r.t. $t$. The partial derivative of the
interpolating quenched pressure with respect to $t$ reads as

$\footnotesize\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
t}&=&\dfrac{1}{N}\mathbb{E}\left[\dfrac{\beta^{\prime}N}{2}\omega(m^{P})\right]+\\\
\\\
&&+\dfrac{\beta^{{}^{\prime}}J}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\sum\limits_{\bm{i}}\mathbb{E}\left[z_{\bm{i}}\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})\right]\,.\end{array}$
(61)

Now, using the Stein’s lemma (21) on standard Gaussian variable $z_{\bm{i}}$

$\footnotesize\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
t}&=&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4N^{{}^{P}}}\sum\limits_{\bm{i}}\left(\mathbb{E}\left[\omega((\sigma_{i_{1}}\cdots\sigma_{i_{P}})^{2})\right]-\mathbb{E}\left[\omega(\sigma_{i_{1}}\cdots\sigma_{i_{P}})^{2}\right]\right)\\\
\\\ &=&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-\langle
q_{12}^{P}\rangle\right)\,.\end{array}$ (62)

∎

######  Proposition 3.

The interpolating pressure (57) at finite size obeys the following
differential equation:

$\frac{d\mathcal{A}^{(P)}_{N}}{dt}=\dfrac{\partial\mathcal{A}_{N}^{(P)}}{\partial
t}+\dot{x}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
x}+\dot{w}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
w}=S(t,\bm{r})+V_{N}(t,\bm{r}),$ (63)

where we set

$\begin{array}[]{lll}\dot{x}=-\dfrac{P}{2}\beta^{\prime}\,{}^{2}J^{2}\bar{q}^{P-1}\,,&&\dot{w}=-\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}\end{array}$
(64)

and

$\begin{array}[]{lll}S(t,\bm{r})&\coloneqq&-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right),\end{array}$
(65)
$\begin{array}[]{lll}V_{N}(t,\bm{r})&\coloneqq&\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{m}^{P-k}\langle(\Delta
m)^{k}\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{q}^{P-k}\langle(\Delta q_{12})^{k}\rangle.\end{array}$ (66)

###### Proof.

Starting to evaluate explicitly $\frac{\partial}{\partial t}\mathcal{A}_{N}$
by using ( (59) - (60) and Definition (24)-(25)) we write

$\footnotesize\begin{array}[]{lll}\dfrac{\partial}{\partial
t}\mathcal{A}^{(P)}_{N}=\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\left(\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(m-\bar{m})^{k}\rangle\bar{m}^{P-k}+\bar{m}^{P}(1-P)+P\bar{m}^{P-1}\langle
m\rangle\right)\\\ \\\
+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(q_{12}-\bar{q})^{k}\rangle\bar{q}^{P-k}+\bar{q}^{P}(1-P)+P\bar{q}^{P-1}\langle
q_{12}\rangle\right)\\\ \\\
=V_{N}(t,\bm{r})+S(t,\bm{r})+\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}\langle
m\rangle+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{2}P\bar{q}^{P-1}\dfrac{1}{2}\left(1-\langle
q_{12}\rangle\right)\\\ \\\
=V_{N}(t,\bm{r})+S(t,\bm{r})+\dfrac{\beta^{{}^{\prime}}J_{0}}{2}P\bar{m}^{P-1}\Big{(}\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial
w}\Big{)}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{2}P\bar{q}^{P-1}\Big{(}\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial
x}\Big{)}\end{array}$ (67)

Thus, by placing $\dot{\bm{r}}=(\dot{x},\dot{w})$ as in (64) we reach the
thesis.

∎

######  Remark 10.

Since the RS assumption, which is linked to the fact that in the thermodynamic
limit

$\displaystyle\langle(m-\bar{m})^{k}\rangle$ $\displaystyle=0\>\ \>k\geq 2$
$\displaystyle\langle(q_{12}-\bar{q})^{k}\rangle$ $\displaystyle=0\>\ \>k\geq
2$

we can say that $V_{N}$ defined in (66) vanishes.

######  Proposition 4.

The transport equation associated to the interpolating pressure
$\mathcal{A}_{N}(t,\bm{r})$ in the thermodynamic limit and under the RS
assumption is

$\footnotesize\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
t}-\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{2}P\bar{q}^{P-1}\left(\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
x}\right)-\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}\left(\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
w}\right)=\\\ \
-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right),\end{array}$
(68)

whose solution is given by

$\small\begin{array}[]{lll}\mathcal{A}^{(P)}_{\rm
RS}(t,\bm{r})=&\ln{2}+\left\langle\ln{\cosh{\left[w+\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}t+z\sqrt{x+\dfrac{P}{2}\beta^{\prime}\,{}^{2}J^{2}\bar{q}^{P-1}t}\right]}}\right\rangle_{z}\\\
\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A^{(P)}_{\rm
RS}(t,\bm{r})=}&-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}t+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}t\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right)\,.\end{array}$
(69)

###### Proof.

We can find the transport equation applying Remark (10) in Prop. (3).

We compute the solution using the characteristic method on the transport
equation:

$\displaystyle\mathcal{A}^{(P)}_{RS}(t,\bm{r})=\mathcal{A}^{(P)}_{RS}(0,\bm{r}-\dot{\bm{r}}t)+S(t,\bm{r})t.$
(70)

where $\dot{\bm{r}}=(\dot{x},\dot{w})$. Along the characteristics, the
fictitious motion in the $(t,\bm{r})$ time-space is linear and returns

$\begin{array}[]{lll}x=x_{0}-\dfrac{P}{2}\beta^{\prime}\,{}^{2}J^{2}\bar{q}^{P-1}t&&w=w_{0}-\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}t\end{array}$
(71)

where $\bm{r}_{0}=(x_{0},w_{0})=(x(t=0),w(t=0))$. The Cauchy condition at
$t=0$ is given by a direct computation at finite $N$ as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(0,\bm{r}-\dot{\bm{r}}t)&=&\mathcal{A}^{(P)}(0,\bm{r}_{0})\\\
\\\
&=&\dfrac{1}{N}\mathbb{E}\Bigg{\\{}\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\exp{}\Bigg{[}w_{0}N\psi\,m(\bm{\sigma})+\sqrt{x_{0}}\displaystyle\sum\limits_{i=1}^{N}z_{i}\sigma_{i}\Bigg{]}\Bigg{\\}}\\\
\\\
&=&\ln{2}+\left\langle\ln{\cosh{\left[\omega_{0}+z\sqrt{y_{0}}\right]}}\right\rangle_{z}\,.\end{array}$
(72)

∎

Giving the suitable values of parameters we have the following

######  Corollary 1.

The RS approximation of the quenched pressure for the P-spin glass model is
obtained by posing $t=1$ and $\bm{r}=\bm{0}$ in (69), which returns

$\begin{array}[]{lll}\mathcal{A}^{(P)}_{\rm
RS}(t,\bm{r})=&\ln{2}+\left\langle\ln{\cosh{\left[\dfrac{P}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P-1}+z\sqrt{\dfrac{P}{2}\beta^{\prime}\,{}^{2}J^{2}\bar{q}^{P-1}}\right]}}\right\rangle_{z}\\\
\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A^{(P)}_{\rm
RS}(t,\bm{r})=}&-\dfrac{P-1}{2}\beta^{{}^{\prime}}J_{0}\bar{m}^{P}+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left(1-P\bar{q}^{P-1}+(P-1)\bar{q}^{P}\right)\,.\end{array}$
(73)

where the order parameters are ruled by the following self-consistency
equations

$\begin{array}[]{lll}\bar{m}=\left\langle\tanh{\left[\dfrac{P}{2}\beta
J_{0}\bar{m}^{{}^{P-1}}+z\beta
J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}\right\rangle_{z}\,,\\\ \\\
\bar{q}=\left\langle\tanh{}^{2}{\left[\dfrac{P}{2}\beta
J_{0}\bar{m}^{{}^{P-1}}+z\beta
J\sqrt{\dfrac{P}{2}\bar{q}^{{}^{P-1}}}\right]}\right\rangle_{z}\,.\end{array}$
(74)

We stress that the expression found with transport equation method in
Corollary (1) is the same found with Guerra’s interpolation in Proposition
(1).

####  1.3.2 1-RSB solution

In this subsection we turn to the solution of the P-spin glass model via the
generalized broken-replica interpolating technique, restricting the
description at the first step of RSB.

The definition of two-replica overlap $q$ distribution is the same as Guerra’s
interpolating technique (See eq. (8)).

Following the same route pursued in the previous sections, we need an
interpolating partition function $\mathcal{Z}$ and an interpolating quenched
pressure $\mathcal{A}$, that are defined hereafter.

######  Definition 12.

Given the interpolating parameters $\bm{r}=(x^{(1)},x^{(2)},w),t$ and the
i.i.d. auxiliary fields $\\{z_{i}^{(1)},z_{i}^{(2)}\\}_{i=1,...,N}$, with
$z_{i}^{(1,2)}\sim\mathcal{N}(0,1)$ for $i=1,...,N$, we can write the 1-RSB
interpolating partition function $\mathcal{Z}_{N}(t,\bm{r})$ for the P-spin
glass model (1) recursively, starting by

$\footnotesize\begin{array}[]{lll}\mathcal{Z}^{(P)}_{2}(t,\bm{r})&:=&\displaystyle\sum\limits_{\\{\bm{\sigma}\\}}\exp{}\Bigg{[}t\dfrac{\beta^{{}^{\prime}}J_{0}N}{2}m^{P}(\bm{\sigma})+wN\,m(\bm{\sigma})+\\\
\\\
&&+\sqrt{t}\beta^{{}^{\prime}}J\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{i_{1}\cdots,i_{{}_{P}}}\sigma_{i_{1}}\cdots\sigma_{i_{P}}+\\\
\\\
&&+\displaystyle\sum\limits_{a=1}^{2}\left(\sqrt{x^{(a)}}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{(a)}\sigma_{i}\right)\Bigg{]}.\end{array}$
(75)

Averaging out the fields recursively, we define

$\displaystyle\mathcal{Z}_{1}(t,\bm{r})\coloneqq$
$\displaystyle\mathbb{E}_{2}\left[\mathcal{Z}_{2}(t,\bm{r})^{\theta}\right]^{1/\theta}$
(76) $\displaystyle\mathcal{Z}_{0}(t,\bm{r})\coloneqq$
$\displaystyle\exp\mathbb{E}_{1}\left[\ln\mathcal{Z}_{1}(t,\bm{r})\right]$
(77) $\displaystyle\mathcal{Z}_{N}(t,\bm{r})\coloneqq$
$\displaystyle\mathcal{Z}_{0}(t,\bm{r}),$ (78)

where with $\mathbb{E}_{a}$ we mean the average over the variables
$z_{i}^{(a)}$’s for $a=1,2$, and with $\mathbb{E}_{0}$ we shall denote the
average over the variables $z_{i_{1}\cdots,i_{{}_{P}}}$’s.

The definition of 1RSB interpolating pressure at finite volume $N$ and in the
thermodynamic limit is the same as Guerra’s interpolating technique (10). The
notation for the generalized average are the analogous as Guerra’s
interpolation.

The next step is building a transport equation for the interpolating quenched
pressure, for which we preliminary need to evaluate the related partial
derivatives, as discussed in the next

######  Lemma 4.

The partial derivative of the interpolating quenched pressure with respect to
a generic variable $\rho$ reads as

$\frac{\partial}{\partial\rho}\mathcal{A}_{N}^{(P)}(t,\bm{r})=\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\left[\mathcal{W}_{2}\omega\big{(}\partial_{\rho}\mathcal{B}(\bm{\sigma};t,\bm{r})\big{)}\right].$
(79)

In particular,

$\displaystyle\frac{\partial}{\partial t}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle+\frac{{\beta^{{}^{\prime}}}^{2}J^{2}}{4}\big{(}1-(1-\theta)\langle
q_{12}^{P}\rangle_{2}-\theta\langle q_{12}^{P}\rangle_{1}\big{)}$ (80)
$\displaystyle\frac{\partial}{\partial x^{(1)}}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{1}{2}\big{(}1-(1-\theta)\langle
q_{12}\rangle_{2}-\theta\langle q_{12}\rangle_{1}\big{)}$ (81)
$\displaystyle\frac{\partial}{\partial x^{(2)}}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{1}{2}\big{(}1-(1-\theta)\langle p_{12}\rangle_{2}\big{)}$
(82) $\displaystyle\frac{\partial}{\partial w}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\langle m\rangle$ (83)

###### Proof.

The proof is pretty lengthy and basically requires just standard calculations
similar to RS case, since we omit it. Here we just prove that, in complete
generality

$\displaystyle\frac{\partial}{\partial\rho}\mathcal{A}^{(P)}_{N}(t,\bm{r})=$
$\displaystyle\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\bigg{[}\partial_{\rho}\ln\mathcal{Z}_{1}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\bigg{[}\frac{1}{\theta}\frac{1}{\mathcal{Z}_{1}}\big{[}\mathcal{Z}_{2}^{\theta}\big{]}^{1/\theta-1}\mathbb{E}_{2}\big{[}\partial_{\rho}\mathcal{Z}_{2}^{\theta}\big{]}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\bigg{[}\frac{\mathcal{Z}_{2}^{\theta}}{\mathbb{E}_{2}\mathcal{Z}_{2}^{\theta}}\frac{\partial_{\rho}\mathcal{Z}_{2}}{\mathcal{Z}_{2}}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\bigg{[}\mathcal{W}_{2}\frac{\partial_{\rho}\mathcal{Z}_{2}}{\mathcal{Z}_{2}}\bigg{]}.$
(84)

∎

######  Proposition 5.

The streaming of the 1-RSB interpolating quenched pressure obeys, at finite
volume $N$, a standard transport equation, that reads as

$\displaystyle\frac{d\mathcal{A}^{(P)}}{dt}$
$\displaystyle=\partial_{t}\mathcal{A}^{(P)}+\dot{x}^{(1)}\partial_{x_{1}}\mathcal{A}^{(P)}+\dot{x}^{(2)}\partial_{x_{2}}\mathcal{A}^{(P)}+\dot{w}\partial_{w}\mathcal{A}^{(P)}=S(t,\bm{r})+V_{N}(t,\bm{r})$
(85)

where

$\displaystyle
S(t,\bm{r})\coloneqq\frac{{\beta^{{}^{\prime}}}^{2}J^{2}}{4}\left[1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}-\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\right]-(P-1)\frac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P}\,,$
(86) $\displaystyle
V_{N}(t,\bm{r})\coloneqq\frac{{\beta^{{}^{\prime}}}^{2}J^{2}}{4}\left\\{(\theta-1)\sum_{k=2}^{P}\binom{P}{k}\bar{q}_{2}^{P-k}\langle(\Delta
q_{2})^{k}\rangle_{2}-\theta\sum_{k=2}^{P}\binom{P}{k}\bar{q}_{1}^{P-k}\langle(\Delta
q_{1})^{k}\rangle_{2}\right\\}$
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}V_{N}(t,\bm{r})\coloneqq}+\frac{\beta^{{}^{\prime}}J_{0}}{2}\sum_{k=2}^{P}\binom{P}{k}\langle(\Delta
m)^{k}\rangle\,,$ (87)

where, as usual, we use $\Delta X_{a}=X-\bar{X}_{a}$.

###### Proof.

Similar to the case RS (24-25), we have, for $a=1,2$

$\displaystyle\langle q_{12}^{P}\rangle_{a}$
$\displaystyle=\sum_{k=1}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{q}_{a}^{P-k}\langle(\Delta
q_{a})^{k}\rangle_{a}+\bar{q}_{a}^{P}(1-P)+P\bar{q}_{a}^{P-1}\langle
q_{12}\rangle_{a}\,.$ (88)

Now, starting to evaluate explicitly $\frac{\partial}{\partial
t}\mathcal{A}^{(P)}_{N}$ by using (81 \- 83) we write

$\displaystyle\frac{\partial}{\partial t}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{\beta^{{}^{\prime}}J_{0}}{2}\left[\sum_{k=2}^{P}\binom{P}{K}\langle(\Delta
m)^{k}\rangle\bar{m}^{P-k}+\bar{m}^{P}(1-P)+P\bar{m}^{P-1}\langle
m\rangle\right]$
$\displaystyle+\frac{{\beta^{{}^{\prime}}}^{2}J^{2}}{4}\Bigg{\\{}1+(\theta-1)\left[\sum_{k=1}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{q}_{2}^{P-k}\langle(\Delta
q_{2})^{k}\rangle_{2}+\bar{q}_{2}^{P}(1-P)+P\bar{q}_{2}^{P-1}\langle
q_{12}\rangle_{2}\right]$
$\displaystyle-\theta\left[\sum_{k=1}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{q}_{1}^{P-k}\langle(\Delta
q_{1})^{k}\rangle_{a}+\bar{q}_{1}^{P}(1-P)+P\bar{q}_{1}^{P-1}\langle
q_{12}\rangle_{1}\right]\Bigg{\\}}=$ $\displaystyle=$ $\displaystyle
V_{N}(t,\bm{r})+S(t,\bm{r})+\frac{\beta^{{}^{\prime}}J_{0}P}{2}\bar{m}^{P-1}\partial_{w}\mathcal{A}^{(P)}_{N}+\frac{\beta^{\prime}\,{}^{2}J^{2}P}{2}\bar{q}_{1}^{P-1}\partial_{x^{(1)}}\mathcal{A}^{(P)}_{N}$
$\displaystyle+\frac{\beta^{\prime}\,{}^{2}J^{2}P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\partial_{x^{(2)}}\mathcal{A}^{(P)}_{N}\,.$
(89)

Thus, by placing

$\displaystyle\dot{x}^{(1)}$
$\displaystyle=-\frac{\beta^{\prime}\,{}^{2}J^{2}P}{2}\bar{q}_{1}^{P-1}\,,$
(90) $\displaystyle\dot{x}^{(2)}$
$\displaystyle=-\frac{\beta^{\prime}\,{}^{2}J^{2}P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\,,$
(91) $\displaystyle\dot{w}$
$\displaystyle=-\frac{\beta^{{}^{\prime}}J_{0}P}{2}\bar{m}^{P-1}\,.$ (92)

thus, we reach the thesis.

∎

######  Remark 11.

In the thermodynamic limit, in the 1RSB scenario, we have

$\displaystyle\lim_{N\rightarrow\infty}\langle(m-\bar{m})^{2}\rangle=$
$\displaystyle 0\,,$ (93)
$\displaystyle\lim_{N\rightarrow\infty}\langle(q_{12}-\bar{q}_{i})^{2}\rangle_{i}=$
$\displaystyle 0;\>\>\>i=1,2\,.$ (94)

Similar to the RS approximation, in the thermodynamic limit we have that the
central moments greater than 2 tend to zero; in this way

$\lim_{N\to\infty}V_{N}(t,\bm{r})=0.$ (95)

Exploiting Remark 11 we can prove the following

######  Proposition 6.

The transport equation associated to the interpolating pressure function
$\mathcal{A}^{(P)}(t,\bm{r})$, in the thermodynamic limit and under the 1RSB
assumption, can be written as

$\displaystyle\partial_{t}\mathcal{A}^{(P)}+\dot{x}^{(1)}\partial_{x_{1}}\mathcal{A}^{(P)}+\dot{x}^{(2)}\partial_{x_{2}}\mathcal{A}^{(P)}+\dot{z}\partial_{z}\mathcal{A}^{(P)}+\dot{w}\partial_{w}\mathcal{A}^{(P)}$
$\displaystyle=\frac{\beta^{\prime}\,{}^{2}J^{2}}{4}\Bigg{[}1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}-\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\Bigg{]}-\frac{\beta^{{}^{\prime}}J_{0}\bar{m}^{P}(P-1)}{2}$
(96)

whose explicit solution is given by

$\displaystyle\mathcal{A}^{(P)}=\log
2+\frac{1}{\theta}\mathbb{E}_{1}\log\mathbb{E}_{2}\cosh^{\theta}\left(w_{0}+z^{(1)}\sqrt{x_{0}^{(1)}}+z^{(2)}\sqrt{x_{0}^{(2)}}\right)$
$\displaystyle+t\left\\{\frac{\beta^{\prime}\,{}^{2}J^{2}}{4}\Bigg{[}1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}-\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\Bigg{]}-\frac{\beta^{{}^{\prime}}J_{0}\bar{m}^{P}(P-1)}{2}\right\\}$
(97)

###### Proof.

If we consider (95) we have (6) We compute the transport equation trough
characteristic method:

$\mathcal{A}^{(P)}_{\textrm{1RSB}}(t,\bm{r})=\mathcal{A}^{(P)}_{\textrm{1RSB}}(0,\bm{r}-\bm{\dot{r}}t)+S_{\textrm{1RSB}}(t,\bm{r})t.$
(98)

By putting (90)-(92) into (85) we find

$\displaystyle\mathcal{A}^{(P)}(t,\bm{r})=$
$\displaystyle\mathcal{A}^{(P)}(0,\bm{r}_{0})+t\left\\{-\frac{\beta^{{}^{\prime}}J_{0}\bar{m}^{P}(P-1)}{2}\right.$
$\displaystyle\left.+\frac{\beta^{\prime}\,{}^{2}J^{2}}{4}\Bigg{[}1-P\bar{q}_{2}^{P-1}+(P-1)\bar{q}_{2}^{P}-\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\Bigg{]}\right\\}$
(99)

where $r_{0}$ can be obtained by using the equation of motion

$\bm{r}=\bm{r}_{0}+\dot{\bm{r}}t$ (100)

where the velocities are defined in (90)-(92). Then, all we have to compute is
$\mathcal{A}^{(P)}_{0}(0,\bm{r}_{0})$, that can be easily done because at
$t=0$ the two body interaction vanishes and it can be written as

$\displaystyle\mathcal{A}^{(P)}_{0}(0,\bm{r}_{0})=$ $\displaystyle\log
2+\frac{1}{\theta}\mathbb{E}_{1}\log\mathbb{E}_{2}\cosh^{\theta}\left(w_{0}+z^{(1)}\sqrt{x_{0}^{(1)}}+z^{(2)}\sqrt{x_{0}^{(2)}}\right).$
(101)

We omit the computation since it is similar to one-body term in Guerra’s
interpolating scheme, 1RSB assumption.
Then, putting together (1.3.2)-(101) and (90)-(92), we finally achieve an
explicit expression for the interpolating pressure of the P-spin glass model
in the 1RSB approximation. ∎

To sum up, we have the following main theorem for the 1RSB scenario

######  Theorem 2.

The 1-RSB quenched pressure for P-spin glass model, in the thermodynamic
limit, reads as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(\beta)=&\log
2+\dfrac{1}{\theta}\mathbb{E}_{1}\log\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})-(P-1)\dfrac{\beta^{{}^{\prime}}J_{0}}{2}\bar{m}^{P}\\\
\\\
&+\dfrac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1+(\theta-1)\bar{q}_{2}^{P}(1-P)-\theta\bar{q}_{1}^{P}(1-P)-P\bar{q}_{2}^{P-1}\right]\end{array}$
(102)

where

$\displaystyle\small g(\bm{z},\bar{m})$
$\displaystyle=\beta^{{}^{\prime}}J_{0}\dfrac{P}{2}\bar{m}^{P-1}+\beta^{{}^{\prime}}Jz^{(1)}\sqrt{\frac{P}{2}\bar{q}_{1}^{P-1}}+\beta^{{}^{\prime}}Jz^{(2)}\sqrt{\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\,.$
(103)

The order parameters are ruled by

$\displaystyle\bar{m}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh
g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]$
(104)
$\displaystyle\bar{q}_{1}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh
g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]^{2}$
(105)
$\displaystyle\bar{q}_{2}=\mathbb{E}_{1}\left[\frac{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})\tanh^{2}g(\bm{z},\bar{m})}{\mathbb{E}_{2}\cosh^{\theta}g(\bm{z},\bar{m})}\right]$
(106)

with $g(\bm{z},\bar{m})$ is the same defined in (103).

###### Proof.

By taking $\bm{r}=\bm{0}$ and $t=1$ we find the P-spin glass pressure in the
1RSB approximation. ∎

We stress that the expression in (102) is the same found with Guerra’s
interpolating scheme in 1RSB assumption.

##  2 Gaussian P-Spin Glass

In this section we face with Gaussian P-spin glass model, namely spin glass
model with Gaussian spins with $P$-wise interactions. Several works describe
the principal aspects of this model, either in our assumption BenArous ;
Bovier3  or in the so-called spherical version, where the Gaussian spin are on
a spherical surfacecrisanti ; PRL ; TalaP . We stress that two assumptions on
Gaussian spins are equivalent, as shown in AdrianoGauss . We deepen the model
through Guerra’s interpolating technique and transport equation for RS and
1RSB assumptions.

###  2.1 Generalities

######  Definition 13.

Let $P\in\mathbb{N}$ and $z_{i}\in\mathbb{R},\;i=1,2,\cdots,N$ be a
configuration of N gaussian spins, the Hamiltonian of the P-spin Gaussian
model is defined as

$H_{N}^{(P)}(\bm{z}|\bm{J})\coloneqq-\dfrac{1}{P!}\displaystyle\sum\limits_{i_{1},\cdots,i_{P}=1}^{N,\cdots,N}\,\mathcal{J}_{i_{1},\cdots,i_{P}}z_{i_{1}}\cdots
z_{i_{P}}$ (107)

where the P-wise quenched couplings
$\bm{\mathcal{J}}=\\{\mathcal{J}_{i_{1},\cdots,i_{P}}\\}_{i_{1},\cdots,i_{P}=1,\cdots,N}$
are given by

$\mathcal{J}_{i_{1},...,i_{P}}\coloneqq\,\sqrt{\dfrac{2}{N^{P-1}}}J_{i_{1},\cdots,i_{P}}$
(108)

with $J_{i_{1},\cdots,i_{P}}$ i.i.d. standard random variables drawn from
$P(J_{i_{1},\cdots,i_{P}})=N(0;1)$.

######  Definition 14.

The partition function related to the Hamiltonian (107) is given by

$\begin{array}[]{lll}Z_{N}(\beta,\bm{J})&\coloneqq\displaystyle\int
d\mu(z)\exp\left[-\beta H_{N}^{(P)}(z|\bm{J})\right]=\mathbb{E}_{z}\,e^{-\beta
H_{N}^{(P)}(z|\bm{J})}\,,\end{array}$ (109)

where, $d\mu(z)=\prod_{i}(2\pi)^{-1/2}e^{-\sum_{i}z_{i}^{2}/2}$ and
$\beta\in\mathbb{R}^{+}$ is the inverse temperature in proper units.

As early pointed out for instance in Berlin , unfortunately these kind of
models need to be regularized; in fact, the right side of (109) is not always
well defined as the P-wise interactions bridges soft spins which are both
Gaussian distributed. It will be clear soon that a good definition is

$\begin{array}[]{lll}Z_{N}(\beta,\bm{J},\lambda)&\coloneqq\mathbb{E}_{z}\,\exp\left[-\beta
H_{N}^{(P)}\left(z|\bm{J}\right)-\left(\dfrac{\beta}{P!}\right)^{2}\dfrac{1}{N^{P-1}}\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}+\dfrac{\lambda}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right]\,,\end{array}$
(110)

where the first additional term is needed for convergence of the integral over
the Gaussian measure $d\mu(z)$. Instead, the new parameter $\lambda$, within
the last term of (110), is just to modify the variance of the soft spins, as
in several applications this can sensibly vary.

We introduce the _Boltzmann average_ induced by the partition function (109),
denoted with $\omega_{\bm{J}}$ and, for an arbitrary observable
$O(\bm{\sigma})$, defined as

$\omega_{\bm{J}}(O(\bm{\sigma})):=\frac{\mathbb{E}_{z}O(\bm{\sigma})e^{-\beta
H_{N}(\bm{\sigma}|\bm{J})}}{Z_{N}(\beta,\bm{J},\lambda)}.$ (111)

This can be further averaged over the realization of the
$J_{i_{1},\cdots,i_{P}}$’s (also referred to as _quenched average_) to get

$\langle
O(\bm{\sigma})\rangle\coloneqq\mathbb{E}\omega_{\bm{J}}(O(\bm{\sigma})).$
(112)

######  Definition 15.

The intensive quenched pressure of the Gaussian P-spin glass model (107) is
defined as

$\mathcal{A}_{N}(\beta,\lambda)\coloneqq\frac{1}{N}\mathbb{E}\ln\mathcal{Z}_{N}(\beta,\bm{J},\lambda),$
(113)

and its thermodynamic limit, assuming its existence, is referred to as

$\mathcal{A}(\beta,\lambda)\coloneqq\lim_{N\to\infty}\mathcal{A}_{N}(\beta,\lambda).$
(114)

In order to solve the model we want to find out an explicit expression for the
quenched pressure (114) in terms of the natural order parameter of the theory,
namely the two-replica overlap $q_{12}$, defined in the following

######  Definition 16.

The order parameter used to describe the macroscopic behavior of the model is
the two-replica overlap, introduced as

$q_{12}\coloneqq\frac{1}{N}\sum_{i=1}^{N}z_{i}^{(1)}z_{i}^{(2)}\,.$ (115)

###  2.2 Resolution via Guerra’s interpolation

Mirroring P-spin glass model, the purpose of this section is to solve the
Gaussian P-spin glass model via Guerra’s interpolating technique. To do so, we
recover the expressions of statistical pressure (which is equivalent to free
energy) and self consistency equations in the approximation of replica
symmetry (RS) and first step of replica symmetry breaking (RSB).

####  2.2.1 RS solution

Using the same definition for $q_{12}$ presented for the P-spin glass model
(def. 5), we can introduce the Guerra’s interpolating partition function of
the Gaussian P-spin glass model.

######  Definition 17.

Given the interpolating parameter $t\in[0,1]$, $A,\ B\in\mathbb{R}$ and
$\tilde{J}_{i}\sim\mathcal{N}(0,1)$ for $i=1,\ldots,N$ standard Gaussian
variables i.i.d., the Guerra’s interpolating partition function is given as

$\footnotesize\begin{array}[]{lll}\mathcal{Z}^{(P)}_{N}(t)&\coloneqq&\mathbb{E}_{z}\exp{}\Bigg{[}\beta^{{}^{\prime}}\sqrt{t}\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{P}=1}^{N,\cdots,N}J_{i_{1},\cdots,i_{P}}z_{i_{1}}\cdots
z_{i_{P}}-t\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P-1}}\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}\\\
\\\
&&+A\sqrt{1-t}\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}z_{i}+(1-t)\dfrac{B}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}+\dfrac{\lambda}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\Bigg{]}\,,\end{array}$
(116)

where $\beta^{{}^{\prime}}=2\beta/P!$.

######  Definition 18.

The interpolating pressure for the Gaussian P-spin glass model (107), at
finite $N$, is introduced as

$\displaystyle\mathcal{A}^{(P)}_{N}(t)$ $\displaystyle\coloneqq$
$\displaystyle\frac{1}{N}\mathbb{E}\left[\ln\mathcal{Z}^{(P)}_{N}(t)\right],$
(117)

where the expectation $\mathbb{E}$ is now meant over $J_{i_{1},\cdots,i_{P}}$
and $\tilde{J}_{i}$, in the thermodynamic limit,

$\mathcal{A}^{(P)}(t)\coloneqq\lim_{N\to\infty}\mathcal{A}^{(P)}_{N}(t).$
(118)

By setting $t=1$ the interpolating pressure recovers the original one (113),
that is $\mathcal{A}_{N}^{(P)}(\beta,\lambda)=\mathcal{A}^{(P)}_{N}(t=1)$.

######  Remark 12.

The interpolating structure implies an interpolating measure, whose related
Boltzmann factor reads as

$\mathcal{B}(\bm{z};t)\coloneqq\exp\left[\beta H(\bm{z};t)\right];$ (119)

In this way the partition function is written as
$\mathcal{Z}_{N}(t)=\mathbb{E}_{z}\mathcal{B}(\bm{z};t)$ .
A generalized average follows from this generalized measure as

$\omega_{t}(O(\bm{z}))\coloneqq\mathbb{E}_{z}O(\bm{z})\mathcal{B}(\bm{z};t)$
(120)

and

$\langle O(\bm{z})\rangle_{t}\coloneqq\mathbb{E}[\omega_{t}(O(\bm{z}))].$
(121)

where $\mathbb{E}$ denotes the average over $\tilde{J}_{i_{1},\cdots,i_{P}}$
and $\\{J_{i}\\}_{i=1,\cdots,N}$.

Of course, when $t=1$ the standard Boltzmann measure and related average is
recovered. Hereafter, in order to lighten the notation, we will drop the
subindices $t$.

######  Lemma 5.

The $t$ derivative of interpolating pressure is given by

$\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}\coloneqq&-\dfrac{\beta^{\prime}\,{}^{2}}{4}\left(\langle
q_{12}\rangle^{P}-\dfrac{2A^{2}}{\beta^{\prime 2}}\langle
q_{12}\rangle\right)-\dfrac{1}{2N}\left(A^{2}+B\right)\displaystyle\sum\limits_{i=1}^{N}\langle
z_{i}^{2}\rangle\end{array}$ (122)

###### Proof.

Deriving equation (117) with respect to $t$, we get

$\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}&=&\dfrac{\beta^{{}^{\prime}}}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}J_{\bm{i}}\omega(z_{i_{1}}\cdots
z_{i_{P}})\right]-\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P}}\left[\sum\limits_{i}\omega(z_{i}^{2})\right]^{P}+\\\
\\\
&&-\dfrac{1}{2N\sqrt{1-t}}\mathbb{E}\left[A\sum\limits_{i}\tilde{J}_{i}\omega(z_{i})\right]-\dfrac{1}{2N}\mathbb{E}\left[B\sum\limits_{i}\omega(z_{i}^{2})\right]\,.\end{array}$
(123)

Now, using the Stein’s lemma (also known as Wick’s theorem) on standard
Gaussian variable $\tilde{J}_{i}$ and $J_{\bm{i}}$, we may rewrite the second
and the third member of (123) as

$\small\begin{array}[]{lll}\dfrac{\beta^{{}^{\prime}}}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}J_{\bm{i}}\omega(z_{i_{1}}\cdots
z_{i_{P}})\right]-\dfrac{1}{2N\sqrt{1-t}}\mathbb{E}\left[A\sum\limits_{i}\tilde{J}_{i}\omega(z_{i})\right]=D_{1}+D_{2}\,.\end{array}$

Let’s investigate those three terms:

$\small\begin{array}[]{lll}D_{1}&=\dfrac{\beta^{{}^{\prime}}}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\mathbb{E}\left[\sum\limits_{\bm{i}}\partial_{J_{\bm{i}}}\omega(z_{i_{1}}\cdots
z_{i_{P}})\right]\\\ \\\
&=\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P}}\left(\sum\limits_{\bm{i}}\mathbb{E}\left[\omega((z_{i_{1}}\cdots
z_{i_{P}})^{2})\right]-\sum\limits_{\bm{i}}\mathbb{E}\left[\omega(z_{i_{1}}\cdots
z_{i_{P}})^{2}\right]\right)\\\ \\\
&=\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P}}\left(\sum\limits_{i}\mathbb{E}\left[\omega(z_{i}^{2})\right]\right)^{P}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\langle
q_{12}^{P}\rangle\,;\end{array}$ (124)
$\footnotesize\begin{array}[]{lll}D_{2}&=-\dfrac{1}{2N\sqrt{1-t}}\mathbb{E}\left[A\sum\limits_{i}\partial_{\tilde{J}_{i}}\omega(z_{i})\right]\\\
\\\
&=-\dfrac{1}{2N}A^{2}\left(\sum\limits_{i}\mathbb{E}\left[\omega(z^{2}_{i})\right]-\sum\limits_{i}\mathbb{E}\left[\omega(z_{i})^{2}\right]\right)\\\
\\\
&=-\dfrac{1}{2}A^{2}\left[\dfrac{1}{N}\mathbb{E}\left[\omega(z_{i})^{2}\right]-\langle
q_{12}\rangle\right]\,.\end{array}$ (125)

Rearranging together (124) and (125) we obtain the thesis. ∎

######  Remark 13.

We stress that, for the RS assumption presented in Definition (5), we can use
the relations (24) and (25) to fix the two constants as

$\begin{array}[]{lll}A^{2}=\dfrac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{{}^{P-1}}\,,\\\
\\\ B=-A^{2}\end{array}$ (126)

the (122) in the thermodynamical limit reads as

$\begin{array}[]{lll}\dfrac{d\mathcal{A}^{(P)}(t)}{dt}&\coloneqq&(P-1)\dfrac{\beta^{\prime}\,{}^{2}}{4}\bar{q}^{P}\end{array}$
(127)

which is now independent of $t$.

######  Theorem 3.

In the thermodynamic limit ($N\to\infty$) and under RS assumption, applying
the Fundamental Theorem of Calculus and using the suitable values of $A$ and
$B$, we find the quenched pressure for the Gaussian P-spin glass model as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(\beta^{{}^{\prime}},\lambda)&=&\dfrac{\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}}{2\left(1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right)}-\dfrac{1}{2}\ln{\left[1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right]}+\\\
\\\ &&+(P-1)\dfrac{\beta^{\prime}\,{}^{2}}{4}\bar{q}^{P}\,.\end{array}$ (128)

where the self-consistency equation which rule the order parameter are given
by the resolution or the implicit equation

$\begin{array}[]{lll}\bar{q}=\dfrac{\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}}{\left(1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right)^{2}}\,.\end{array}$
(129)

###### Proof.

Using the Fundamental Theorem of Calculus (30) and computing the one-body
terms

$\footnotesize\begin{array}[]{lll}\mathcal{A}^{(P)}(t=0)&=&\dfrac{1}{N}\mathbb{E}_{\tilde{J}}\ln{}\Bigg{\\{}\mathbb{E}_{z}\exp{}\Bigg{[}A\,\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}z_{i}+\dfrac{1}{2}\left(B+\lambda\right)\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\Bigg{]}\Bigg{\\}}\,,\\\
\\\
&=&\dfrac{A^{2}}{2\left(1-\lambda-B\right)}-\dfrac{1}{2}\ln{\left[1-\lambda-B\right]}\\\
\\\
&=&\dfrac{\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}}{2\left(1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right)}-\dfrac{1}{2}\ln{\left[1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right]}\,.\end{array}$
(130)

Finally, putting (127) and (130) in (30), we find (128).

Extremizing the statistical pressure in (128) w.r.t. the order parameter we
find equation (129). ∎

A, briefly, deeper study of the interesting case of $P=2$ is done in Section
2.4, for a complete discussion we remand to AdrianoGauss .

######  Remark 14.

We highlight that we recover the same results in crisanti  in RS assumption
computed via replica trick.

Without going into details of implicit equation (129), if we focus on $P>2$
case, we stress that the solution $\bar{q}=0$ is always a saddle point; for
$\bar{q}\neq 0$ we can see numerically that there is only an acceptable
solutions, which maximize the quenched statistical pressure. For every details
we remind to crisanti .

####  2.2.2 1RSB solution

In this subsection we turn to the solution of the Gaussian P-spin glass model
via the Guerra’s interpolating technique, restricting the description at the
first step of RSB.

Following the same route pursued in the previous sections, we need an
interpolating partition function $\mathcal{Z}$ and an interpolating quenched
pressure $\mathcal{A}$, that are defined hereafter.

######  Definition 19.

Given the interpolating parameter $t$ and the i.i.d. auxiliary fields
$\\{\tilde{J}_{i}^{(1)},\tilde{J}_{i}^{(2)}\\}_{i=1,...,N}$, with
$\tilde{J}_{i}^{(1,2)}\sim\mathcal{N}(0,1)$ for $i=1,...,N$ we can write the
1-RSB interpolating partition function $\mathcal{Z}_{N}(t)$ for the P-spin
Gaussian model (110) recursively, starting by

$\begin{array}[]{lll}\mathcal{Z}^{(P)}_{2}(t)&:=&\mathbb{E}_{z}\exp\Bigg{[}\sqrt{t}\beta^{{}^{\prime}}\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}J_{i_{1}\cdots,i_{{}_{P}}}z_{i_{1}}\cdots
z_{i_{P}}-t\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P-1}}\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}\\\
\\\
&&+\sqrt{1-t}\displaystyle\sum\limits_{a=1}^{2}\left(A^{(a)}\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}^{(a)}z_{i}\right)+(1-t)\dfrac{B}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}+\dfrac{\lambda}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\Bigg{]}\end{array}$
(131)

where the $J_{i_{1}\cdots,i_{{}_{P}}}$’s i.i.d. standard Gaussian. Averaging
out the fields recursively, we define

$\displaystyle\mathcal{Z}_{1}^{(P)}(t)\coloneqq$
$\displaystyle\mathbb{E}_{2}\left[\mathcal{Z}_{2}^{(P)}(t)^{\theta}\right]^{1/\theta}$
(132) $\displaystyle\mathcal{Z}_{0}^{(P)}(t)\coloneqq$
$\displaystyle\exp\mathbb{E}_{1}\left[\ln\mathcal{Z}_{1}^{(P)}(t)\right]$
(133) $\displaystyle\mathcal{Z}_{N}^{(P)}(t)\coloneqq$
$\displaystyle\mathcal{Z}_{0}^{(P)}(t),$ (134)

where with $\mathbb{E}_{a}$ we mean the average over the variables
$\tilde{J}_{i}^{(a)}$’s, for $a=1,2$, and with $\mathbb{E}_{0}$ we shall
denote the average over the variables $J_{i_{1}\cdots,i_{{}_{P}}}$’s.

######  Definition 20.

The 1RSB interpolating pressure, at finite volume $N$, is introduced as

$\mathcal{A}_{N}^{(P)}(t)\coloneqq\frac{1}{N}\mathbb{E}_{0}\big{[}\ln\mathcal{Z}_{0}^{(P)}(t)\big{]},$
(135)

and, in the thermodynamic limit, assuming its existence

$\mathcal{A}^{(P)}(t)\coloneqq\lim_{N\to\infty}\mathcal{A}^{(P)}_{N}(t).$
(136)

By setting $t=1$, the interpolating pressure recovers the standard pressure
(113), that is,
$\mathcal{A}^{(P)}_{N}(\beta,\lambda)=\mathcal{A}^{(P)}_{N}(t=1)$.

Now the next step is computing the $t$-derivative of the statistical pressure.
In this way we can apply the fundamental theorem of calculus and find the
solution of the original model.

######  Lemma 6.

The derivative w.r.t. $t$ of interpolating statistical pressure can be written
as

$\begin{array}[]{lll}d_{t}\mathcal{A}^{(P)}_{N}=&+\dfrac{\beta^{\prime}\,{}^{2}}{4}\left[(\theta-1)\langle
q_{12}^{P}\rangle_{2}-\theta\langle
q_{12}^{P}\rangle_{1}\right]-\frac{(A^{(1)})^{2}}{2}\left[\frac{1}{N}\sum_{i}\langle
z_{i}^{2}\rangle+(\theta-1)\langle q_{12}\rangle_{2}-\theta\langle
q_{12}\rangle_{1}\right]\\\ \\\
&-\frac{(A^{(2)})^{2}}{2}\left[\frac{1}{N}\sum_{i}\langle
z_{i}^{2}\rangle+(\theta-1)\langle
q_{12}\rangle_{2}\right]-\frac{B}{2N}\sum_{i}\langle
z_{i}^{2}\rangle\end{array}$ (137)

Since the proof is similar to SK spin glass case, we omit it.

######  Remark 15.

If we use (88), in 1RSB assumption we fix the contasts as

$\displaystyle A_{1}$
$\displaystyle=\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{1}^{P-1}\,,$ (138)
$\displaystyle A_{2}$
$\displaystyle=\beta^{\prime}\,{}^{2}\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\,,$
(139) $\displaystyle B$
$\displaystyle=-\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\,.$ (140)

Thus, the derivative w.r.t. $t$ in the thermodynamic limit is computed as

$\displaystyle
d_{t}\mathcal{A}^{(P)}_{N}=\frac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\frac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\,.$
(141)

######  Proposition 7.

At finite size and under 1RSB assumption applying the Fundamental Theorem of
Calculus and using the suitable values of $A^{(1)},A^{(2)},B$, we find the
quenched pressure for the Gaussian P-spin glass model as

$\begin{array}[]{lll}\mathcal{A}^{(P)}=&\dfrac{\beta^{\prime}\,{}^{2}}{4}(\theta-1)\left[\displaystyle\sum\limits_{K=2}^{P}\langle
q_{12}^{k}\rangle_{2}\bar{q}_{2}^{P-k}+(1-P)\bar{q}_{2}^{P}\right]\\\ \\\
&-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta\left[\displaystyle\sum\limits_{k=2}^{P}\langle
q_{12}^{k}\rangle_{1}\bar{q}_{1}^{P-k}+(1-P)\bar{q}_{1}^{P}\right]-\dfrac{1}{2}\log\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right)\\\
\\\
&+\dfrac{1}{2\theta}\log\left(\dfrac{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\frac{P}{2}\theta(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\right)\\\
\\\
&+\dfrac{\beta^{\prime}\,{}^{2}P}{4}\dfrac{\bar{q}_{1}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\end{array}$
(142)

###### Proof.

We use the fundamental theorem of Calculus (30). The last step we need is the
computation of one-body term. Computing the Gaussian integral, we have

$\begin{array}[]{lll}\mathcal{A}^{(P)}(t=0)=&-\dfrac{1}{2}\log\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right)\\\
\\\
&+\dfrac{1}{2\theta}\log\left(\dfrac{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-(\beta^{{}^{\prime}})^{2}\frac{P}{2}\theta(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\right)\\\
\\\
&+\dfrac{\beta^{\prime}\,{}^{2}P}{4}\dfrac{\bar{q}_{1}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\,.\end{array}$
(143)

In this way, we obtain the thesis. ∎

Now we have the following

######  Theorem 4.

In the thermodynamic limit, under 1RSB assumption, the expression of quenched
statistical pressure for Gaussian P-spin glass model (110) is

$\begin{array}[]{lll}\mathcal{A}^{(P)}(\beta,\lambda)=&\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\\\
\\\
&-\dfrac{1}{2}\log\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right)\\\
\\\
&+\dfrac{1}{2\theta}\log\left(\dfrac{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-(\beta^{{}^{\prime}})^{2}\frac{P}{2}\theta(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\right)\\\
\\\
&+\dfrac{\beta^{\prime}\,{}^{2}P}{4}\dfrac{\bar{q}_{1}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})}\end{array}$
(144)

where the order parameters are ruled by the implicit equations

$\footnotesize\begin{array}[]{lll}\bar{q}_{1}=\dfrac{\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{1}^{P-1}}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\right)^{2}}\\\
\\\
\bar{q}_{2}=\bar{q}_{1}+\dfrac{\beta^{\prime}\,{}^{2}\frac{P}{2}\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)\right)\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right)}\end{array}$
(145)

###### Proof.

If we use (141) and (143) in (30) we recover (144).

Extremizing the statistical pressure in (144) w.r.t. the order parameters we
find the self-consistency equations. ∎

######  Remark 16.

In the case of pairwise interaction ($P=2$) the 1RSB solution coincide with
the RS one, as shown in Section 2.4.

Moreover, we recover the same expression in crisanti ; we remind to it for the
study of solutions behaviour of order parameters (145).

###  2.3 Resolution via transport equation

Now the aim is to show that, using the transport equation technique, already
introduced in Section 1.3 for the RS assumption and for 1RSB in P-spin glass
network, we are able to retrieve the same expression of statistical pressure
above found by Guerra’s Interpolation.

####  2.3.1 RS solution

As previous presented in Section 1.3, now we introduce the interpolating
partitions function for the Gaussian P-spin model.

######  Definition 21.

Given the interpolating parameter $t,x,w$, and
$\tilde{J}_{i}\sim\mathcal{N}(0,1)$ standard Gaussian variables iid, the
partition function is given as

$\footnotesize\begin{array}[]{lll}\mathcal{Z}^{(P)}_{N}(t,\bm{r})&\coloneqq&\mathbb{E}_{z}\exp{}\Bigg{[}\beta^{{}^{\prime}}\sqrt{t}\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{P}=1}^{N,\cdots,N}J_{i_{1},\cdots,i_{P}}z_{i_{1}}\cdots
z_{i_{P}}-t\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P-1}}\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}\\\
\\\
&&+\sqrt{x}\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}z_{i}+\dfrac{w}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}+\dfrac{\lambda}{2}\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\Bigg{]}\,,\end{array}$
(146)

where we set $\beta^{\prime}=2\beta/P!$.

Similar to RS Guerra’s interpolation, we can define the interpolating
pressure, the Boltzmann factor and the generalized measure.

######  Lemma 7.

The partial derivatives of the interpolating pressure (117) w.r.t. $t,x,w$
give the following expectation values:

$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial t}$
$\displaystyle=$ $\displaystyle-\dfrac{\beta^{\prime}\,{}^{2}}{4}\langle
q_{12}^{P}\rangle,$ (147)
$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial x}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2N}\displaystyle\sum\limits_{i=1}^{N}\langle
z_{i}^{2}\rangle-\dfrac{1}{2}\langle q_{12}\rangle,$ (148)
$\displaystyle\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial w}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2N}\displaystyle\sum\limits_{i=1}^{N}\langle
z_{i}^{2}\rangle.$ (149)

###### Proof.

We prove only the derivatives w.r.t. $t$, as long as the procedure is
analogous for each parameter. The partial derivative of the interpolating
quenched pressure with respect to $t$ reads as

$\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
t}&=&\dfrac{\beta^{{}^{\prime}}}{2N\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\sum\limits_{\bm{i}}\mathbb{E}\left[\omega(z_{i_{1}}\cdots
z_{i_{P}})\right]-\dfrac{\beta^{\prime
2}}{4N^{P}}\mathbb{E}\left[\omega\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}\right]\,.\end{array}$
(150)

Now, using the Stein’s lemma (21) on standard Gaussian variable
$z_{i_{1},\cdots,i_{P}}$

$\footnotesize\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
t}&=&\dfrac{\beta^{\prime}\,{}^{2}}{4N^{{}^{P}}}\left\\{\mathbb{E}\left[\omega((z_{i_{1}}\cdots
z_{i_{P}})^{2})\right]-\mathbb{E}\left[\omega(z_{i_{1}}\cdots
z_{i_{P}})^{2}\right]-\mathbb{E}\left[\omega\left(\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}\right)^{P}\right]\right\\}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\langle
q_{12}^{P}\rangle\,.\end{array}$ (151)

∎

######  Proposition 8.

The interpolating pressure (146) at finite size obeys the following
differential equation:

$\frac{d\mathcal{A}^{(P)}_{N}}{dt}=\dfrac{\partial\mathcal{A}_{N}^{(P)}}{\partial
t}+\dot{x}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
x}+\dot{w}\dfrac{\partial\mathcal{A}^{(P)}_{N}}{\partial
w}=S(t,\bm{r})+V_{N}(t,\bm{r}),$ (152)

where we set

$\begin{array}[]{lll}\dot{x}=-\dfrac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\,,&&\dot{w}=-\dot{x}\end{array}$
(153)

and

$\begin{array}[]{lll}S(t,\bm{r})&\coloneqq&\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}^{P},\end{array}$
(154)
$\begin{array}[]{lll}V_{N}(t,\bm{r})&\coloneqq&\dfrac{\beta^{\prime}\,{}^{2}}{4}\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{q}^{P-k}\langle(\Delta q_{12})^{k}\rangle.\end{array}$
(155)

###### Proof.

Starting to evaluate explicitly $\frac{\partial}{\partial t}\mathcal{A}_{N}$
by using ( (148) - (149) and Definition (24)-(25)) we write

$\footnotesize\begin{array}[]{lll}\dfrac{\partial}{\partial
t}\mathcal{A}^{(P)}_{N}=-\dfrac{\beta^{\prime}\,{}^{2}}{4}\left(\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\langle(q_{12}-\bar{q})^{k}\rangle\bar{q}^{P-k}-\bar{q}^{P}(1-P)-P\bar{q}^{P-1}\langle
q_{12}\rangle\right)\\\ \\\
=V_{N}(t,\bm{r})+S(t,\bm{r})+\dfrac{\beta^{\prime}\,{}^{2}}{2}P\bar{q}^{P-1}\dfrac{1}{2}\langle
q_{12}\rangle\\\ \\\
=V_{N}(t,\bm{r})+S(t,\bm{r})+\dfrac{\beta^{\prime}\,{}^{2}}{2}P\bar{q}^{P-1}\Big{(}\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial
w}\Big{)}-\dfrac{\beta^{\prime}\,{}^{2}J}{2}P\bar{q}^{P-1}\Big{(}\frac{\partial\mathcal{A}^{(P)}_{N}}{\partial
x}\Big{)}\end{array}$ (156)

Thus, by placing $\dot{\bm{r}}=(\dot{x},\dot{w})$ as in (153) we reach the
thesis. ∎

######  Remark 17.

As shown for the P-spin glass model in the RS assumption (Remark 10), even for
the Gaussian P-spin glass model, we can say that $V_{N}$, defined in (155),
vanishes.

######  Proposition 9.

The transport equation associated to the interpolating pressure
$\mathcal{A}_{N}(t,\bm{r})$ in the thermodynamic limit and under the RS
assumption is

$\footnotesize\begin{array}[]{lll}\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
t}+\dfrac{\beta^{\prime}\,{}^{2}}{2}P\bar{q}^{P-1}\left(\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
x}\right)-\dfrac{\beta^{\prime}\,{}^{2}}{2}P\bar{q}^{P-1}\left(\dfrac{\partial\mathcal{A}^{(P)}_{\textrm{RS}}}{\partial
w}\right)=\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}^{P},\end{array}$ (157)

whose solution is given by

$\small\begin{array}[]{lll}\mathcal{A}^{(P)}_{\rm
RS}(t,\bm{r})=&\frac{1}{2}-\ln{\left[1-\lambda-(w-\frac{P}{2}\beta^{{}^{\prime}}\bar{q}^{P-1}t)\right]}+\dfrac{x+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}t}{2\left(1-\lambda-(w-\frac{P}{2}\beta^{{}^{\prime}}\bar{q}^{P-1}t)\right)}\\\
\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A^{(P)}_{\rm
RS}(t,\bm{r})=}&+\dfrac{\beta^{\prime}\,{}^{2}}{4}t(P-1)\bar{q}^{P}\,.\end{array}$
(158)

###### Proof.

We can find the transport equation applying Remark (17) in Prop. (8).

We compute the solution using the characteristic method on the transport
equation:

$\displaystyle\mathcal{A}^{(P)}_{RS}(t,\bm{r})=\mathcal{A}^{(P)}_{RS}(0,\bm{r}-\dot{\bm{r}}t)+S(t,\bm{r})t.$
(159)

where $\dot{\bm{r}}=(\dot{x},\dot{w})$. Along the characteristics, the
fictitious motion in the $(t,\bm{r})$ time-space is linear and returns

$\begin{array}[]{lll}x=x_{0}-\dfrac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}t&&w=w_{0}+\dfrac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}t\end{array}$
(160)

where $\bm{r}_{0}=(x_{0},w_{0})=(x(t=0),w(t=0))$. The Cauchy condition at
$t=0$ is given by a direct computation at finite $N$ as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(0,\bm{r}-\dot{\bm{r}}t)&=&\mathcal{A}^{(P)}(0,\bm{r}_{0})\\\
\\\
&=&\dfrac{1}{N}\mathbb{E}\Bigg{\\{}\mathbb{E}_{z}\exp{}\Bigg{[}\dfrac{1}{2}\left(w_{0}+\lambda\right)\displaystyle\sum\limits_{i=1}^{N}z_{i}^{2}+\sqrt{x_{0}}\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}z_{i}\Bigg{]}\Bigg{\\}}\\\
\\\ &=&-\frac{1}{2}\ln{\left[1-\lambda-
w_{0}\right]}+\dfrac{x_{0}}{2\left(1-\lambda-w_{0}\right)}\,.\end{array}$
(161)

∎

Giving the suitable values of parameters we have the following

######  Corollary 2.

The RS approximation of the quenched pressure, in the thermodynamic limit, for
the Gaussian P-spin glass model is obtained by posing $t=1$ and
$\bm{r}=\bm{0}$ in (158), which returns

$\begin{array}[]{lll}\mathcal{A}^{(P)}_{\rm
RS}(\beta^{{}^{\prime}},\lambda)=&-\frac{1}{2}\ln{\left[1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right]}+\dfrac{\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}}{2\left(1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right)}\\\
\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}A^{(P)}_{\rm
RS}(t,\bm{r})=}&+\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}^{P}\,.\end{array}$
(162)

where the self-consistency equation which rule the order parameter are

$\begin{array}[]{lll}\bar{q}=\dfrac{\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}}{\left(1-\lambda+\frac{P}{2}\beta^{\prime}\,{}^{2}\bar{q}^{P-1}\right)^{2}}\,.\end{array}$
(163)

We stress that the expression found with transport equation method in Cor. (2)
is the same found with Guerra’s interpolation in Prop. (3).

####  2.3.2 1-RSB solution

In this subsection we turn to the solution of the Gaussian spin glass model
through transport equation via the generalized broken-replica interpolating
technique, restricting the description at the first step of RSB.

The definition of two-replica overlap $q$ distribution is the same as Guerra’s
interpolation (See Def. (8)).

Following the same route pursued in the previous sections, we need an
interpolating partition function $\mathcal{Z}$ and an interpolating quenched
pressure $\mathcal{A}$, that are defined hereafter.

######  Definition 22.

Given the interpolating parameters $\bm{r}=(x^{(1)},x^{(2)},w),t$ and the
i.i.d. auxiliary fields
$\\{\tilde{J}_{i}^{(1)},\tilde{J}_{i}^{(2)}\\}_{i=1,...,N}$, with
$\tilde{J}_{i}^{(1,2)}\sim\mathcal{N}(0,1)$ for $i=1,...,N$, we can write the
1RSB interpolating partition function $\mathcal{Z}_{N}(t,\bm{r})$ for the
Gaussian P-spin glass model (107) recursively, starting by

$\footnotesize\begin{array}[]{lll}\mathcal{Z}^{(P)}_{2}(t,\bm{r})&:=&\mathbb{E}_{z}\exp{}\Bigg{[}\sqrt{t}\beta^{{}^{\prime}}\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}J_{i_{1}\cdots,i_{{}_{P}}}z_{i_{1}}\cdots
z_{i_{P}}-t\dfrac{\beta^{\prime}\,{}^{2}}{4N^{P-1}}\left(\displaystyle\sum\limits_{i}z_{i}^{2}\right)^{P}\\\
\\\
&&+\displaystyle\sum\limits_{a=1}^{2}\left(\sqrt{x^{(a)}}\displaystyle\sum\limits_{i=1}^{N}\tilde{J}_{i}^{(a)}z_{i}\right)+\frac{w}{2}\sum_{i}z_{i}^{2}+\frac{\lambda}{2}\sum_{i}z_{i}^{2}\Bigg{]}.\end{array}$
(164)

Averaging out the fields recursively, we define

$\displaystyle\mathcal{Z}_{1}(t,\bm{r})\coloneqq$
$\displaystyle\mathbb{E}_{2}\left[\mathcal{Z}_{2}(t,\bm{r})^{\theta}\right]^{1/\theta}$
(165) $\displaystyle\mathcal{Z}_{0}(t,\bm{r})\coloneqq$
$\displaystyle\exp\mathbb{E}_{1}\left[\ln\mathcal{Z}_{1}(t,\bm{r})\right]$
(166) $\displaystyle\mathcal{Z}_{N}(t,\bm{r})\coloneqq$
$\displaystyle\mathcal{Z}_{0}(t,\bm{r}),$ (167)

where with $\mathbb{E}_{a}$ we mean the average over the variables
$\tilde{J}_{i}^{(a)}$’s for $a=1,2$, and with $\mathbb{E}_{0}$ we shall denote
the average over the variables $J_{i_{1}\cdots,i_{{}_{P}}}$’s.

The definition of 1RSB interpolating pressure at finite volume $N$ and in the
thermodynamic limit and the notation for the generalized average are the same
as Guerra’s interpolating technique (20).

The next step is building a transport equation for the interpolating quenched
pressure, for which we preliminary need to evaluate the related partial
derivatives, as discussed in the next

######  Lemma 8.

The partial derivative of the interpolating quenched pressure with respect to
each variable is

$\displaystyle\frac{\partial}{\partial t}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{\beta^{\prime}\,{}^{2}}{2}(\theta-1)\langle
q_{12}^{P}\rangle_{2}-\frac{\beta^{\prime}\,{}^{2}}{2}\theta\langle
q_{12}^{P}\rangle_{1}$ (168) $\displaystyle\frac{\partial}{\partial
x^{(1)}}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{1}{2}\left(\frac{1}{N}\sum_{i}\langle
z_{i}^{2}\rangle-(1-\theta)\langle q_{12}\rangle_{2}-\theta\langle
q_{12}\rangle_{1}\right)$ (169) $\displaystyle\frac{\partial}{\partial
x^{(2)}}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{1}{2}\left(\frac{1}{N}\sum_{i}\langle
z_{i}^{2}\rangle-(1-\theta)\langle q_{12}\rangle_{2}\right)$ (170)
$\displaystyle\frac{\partial}{\partial w}\mathcal{A}^{(P)}_{N}=$
$\displaystyle\frac{1}{2N}\sum_{i}\langle z_{i}^{2}\rangle$ (171)

The proof is similar to Guerra’s interpolation technique, since we omit it.

######  Proposition 10.

The streaming of the 1-RSB interpolating quenched pressure obeys, at finite
volume $N$, a standard transport equation, that reads as

$\displaystyle\frac{d\mathcal{A}^{(P)}}{dt}$
$\displaystyle=\partial_{t}\mathcal{A}^{(P)}+\dot{x}^{(1)}\partial_{x_{1}}\mathcal{A}^{(P)}+\dot{x}^{(2)}\partial_{x_{2}}\mathcal{A}^{(P)}+\dot{w}\partial_{w}\mathcal{A}^{(P)}=S_{\textrm{1RSB}}(t,\bm{r})+V_{N}(t,\bm{r})$
(172)

where

$\displaystyle\dot{x}^{(1)}$
$\displaystyle=-\frac{\beta^{\prime}\,{}^{2}P}{2}\bar{q}_{1}^{P-1}$ (173)
$\displaystyle\dot{x}^{(2)}$
$\displaystyle=-\frac{\beta^{\prime}\,{}^{2}P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})$
(174) $\displaystyle\dot{w}$
$\displaystyle=+\frac{\beta^{\prime}\,{}^{2}P}{2}\bar{q}_{2}^{P-1}$ (175)

and

$\displaystyle
S_{\textrm{1RSB}}(t,\bm{r})\coloneqq\frac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\frac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})$
(176) $\displaystyle
V_{N}(t,\bm{r})\coloneqq\frac{\beta^{\prime}\,{}^{2}}{4}(\theta-1)\left(\sum_{k=2}^{P}\langle\Delta
q_{12}^{k}\rangle_{2}\bar{q}_{2}^{P-k}\right)-\frac{\beta^{\prime}\,{}^{2}}{4}\theta\sum_{k=2}^{P}\langle\Delta
q_{12}^{k}\rangle_{1}\bar{q}_{1}^{P-k}$ (177)

###### Proof.

Keeping in mind the expression (88), we start to evaluate explicitly
$\frac{\partial}{\partial t}\mathcal{A}^{(P)}_{N}$ by using (169 \- 171), so
we get

$\displaystyle\frac{\partial}{\partial
t}\mathcal{A}^{(P)}_{N}=\frac{\beta^{\prime}\,{}^{2}}{4}\Bigg{\\{}(\theta-1)\left[\sum_{k=2}^{P}\langle\Delta
q_{12}^{k}\rangle_{2}\bar{q}_{2}^{P-k}+\bar{q}_{2}^{P}(1-P)+P\bar{q}_{2}^{P-1}\langle
q_{12}\rangle_{2}\right]$
$\displaystyle-\theta\left[\sum_{k=2}^{P}\langle\Delta
q_{12}^{k}\rangle_{1}\bar{q}_{1}^{P-k}+\bar{q}_{1}^{P}(1-P)+P\bar{q}_{1}^{P-1}\langle
q_{12}\rangle_{1}\right]\Bigg{\\}}=$
$\displaystyle=V_{N}(t,\bm{r})+S(t,\bm{r})+\frac{\beta^{\prime}\,{}^{2}}{4}P(\theta-1)\bar{q}_{2}^{P-1}\langle
q_{12}\rangle_{2}-\theta(1-P)\bar{q}_{1}^{P}$
$\displaystyle=V_{N}(t,\bm{r})+\frac{\beta^{\prime}\,{}^{2}(\theta-1)}{4}(1-P)\bar{q}_{2}^{P}-\frac{\beta^{\prime}\,{}^{2}\theta}{4}\bar{q}_{1}^{P}+\frac{\beta^{\prime}\,{}^{2}P}{2}\bar{q}_{1}^{P-1}\partial_{x^{(1)}}\mathcal{A}^{(P)}_{N}$
$\displaystyle+\frac{\beta^{\prime}\,{}^{2}P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\partial_{x^{(2)}}\mathcal{A}^{(P)}_{N}-\frac{\beta^{\prime}\,{}^{2}P}{2}\bar{q}_{2}^{P-1}\partial_{w}\mathcal{A}^{(P)}_{N}$
(178)

Thus, using (173)-(175) we reach the thesis.

∎

######  Remark 18.

In the thermodynamic limit, in the 1RSB scenario, we have that the central
moments greater than 2 tend to zero; in this way

$\lim_{N\to\infty}V_{N}(t,\bm{r})=0.$ (179)

Exploiting Remark 18 we can prove the following

######  Proposition 11.

The transport equation associated to the interpolating pressure function
$\mathcal{A}^{(P)}_{N}(t,\bm{r})$, in the thermodynamic limit and under the
1RSB assumption, can be written as

$\displaystyle\partial_{t}\mathcal{A}^{(P)}+\dot{x}^{(1)}\partial_{x_{1}}\mathcal{A}^{(P)}+\dot{x}^{(2)}\partial_{x_{2}}\mathcal{A}^{(P)}+\dot{w}\partial_{w}\mathcal{A}^{(P)}$
$\displaystyle=\frac{\beta^{\prime}\,{}^{2}}{4}(\theta-1)(1-P)\bar{q}_{2}^{P}-\frac{\beta^{\prime}\,{}^{2}}{4}\theta(1-P)\bar{q}_{1}^{P}$
(180)

whose explicit solution is given by

$\small\begin{array}[]{lll}\mathcal{A}^{(P)}_{1RSB}=&t\left[\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\right]-\dfrac{1}{2}\log\left(1-\lambda-
w_{0}\right)\\\ \\\ &+\dfrac{1}{2\theta}\log\left(\dfrac{1-\lambda-
w_{0}}{1-\lambda-w_{0}-\theta
x_{0}^{(2)}}\right)+\dfrac{x_{0}^{(1)}}{2(1-\lambda-w_{0}-\theta
x_{0}^{(2)})}\end{array}$ (181)

###### Proof.

We compute through characteristic method:

$\displaystyle\mathcal{A}^{(P)}_{\textrm{1RSB}}(t,\bm{r})=\mathcal{A}^{(P)}_{\textrm{1RSB}}(0,\bm{r}-\bm{\dot{r}}t)+S_{\textrm{1RSB}}(t,\bm{r})t$
(182)

By putting (173)-(175) into (172) we find

$\small\mathcal{A}^{(P)}_{0}(t,\bm{r})=\mathcal{A}^{(P)}_{0}(0,\bm{r}_{0})+t\left[\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})\right]$
(183)

where $r_{0}$ can be obtained by using the equation of motion

$\bm{r}=\bm{r}_{0}+\dot{\bm{r}}t$ (184)

where the velocities are defined in (173)-(175). Then, all we have to compute
is $\mathcal{A}^{(P)}_{0}(0,\bm{r}_{0})$, that can be easily done because at
$t=0$ the two body interaction vanishes and the (164) can be written as

$\displaystyle\mathcal{A}^{(P)}_{0}(0,\bm{r}_{0})=$
$\displaystyle-\frac{1}{2}\log\left(1-\lambda-
w_{0}\right)+\frac{1}{2\theta}\log\left(\frac{1-\lambda-w_{0}}{1-\lambda-
w_{0}-\theta x_{0}^{(2)}}\right)$
$\displaystyle+\frac{x_{0}^{(1)}}{2(1-\lambda-w_{0}-\theta x_{0}^{(2)})}.$
(185)

We omit the computation since it is similar to one-body term in Guerra’s
interpolating scheme, 1RSB assumption.
Then, putting together (183)-(2.3.2) and (173)-(175), we finally achieve an
explicit expression for the interpolating pressure of the Gaussian P-spin
glass model in the 1RSB approximation. ∎

To sum up, we have the following main theorem for the 1RSB scenario.

######  Theorem 5.

The 1-RSB quenched pressure for Gaussian P-spin glass model, in the
thermodynamic limit, reads as

$\begin{array}[]{lll}\mathcal{A}^{(P)}(\beta^{{}^{\prime}},\lambda)=&\dfrac{\beta^{\prime}\,{}^{2}}{4}(P-1)\bar{q}_{2}^{P}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta(P-1)(\bar{q}_{2}^{P}-\bar{q}_{1}^{P})-\frac{1}{2}\log\left[1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right]\\\
\\\
&+\dfrac{1}{2\theta}\log\left[\dfrac{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\frac{P}{2}\theta\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)}\right]\\\
\\\
&+\dfrac{\beta^{\prime}\,{}^{2}\frac{P}{4}\bar{q}_{1}^{P-1}}{1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\frac{P}{2}\theta\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)}\end{array}$
(186)

where the order parameters are ruled by

$\begin{array}[]{lll}\bar{q}_{1}=\dfrac{\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{1}^{P-1}}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1})\right)^{2}}\\\
\\\
\bar{q}_{2}=\bar{q}_{1}+\dfrac{\beta^{\prime}\,{}^{2}\frac{P}{2}\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}-\beta^{\prime}\,{}^{2}\theta\frac{P}{2}\left(\bar{q}_{2}^{P-1}-\bar{q}_{1}^{P-1}\right)\right)\left(1-\lambda+\beta^{\prime}\,{}^{2}\frac{P}{2}\bar{q}_{2}^{P-1}\right)}\end{array}$
(187)

###### Proof.

By taking $\bm{r}=\bm{0}$ and $t=1$ we find the P-spin Gaussian pressure in
the 1RSB approximation. Extremizing (186) we obtain the self-consistency
equations. ∎

We stress that the expression in (186) is the same found with Guerra’s
interpolating scheme (144).

###  2.4 A deeper analysis of Gaussian spin glass model

In this section we will focus more specifically on the case of $P=2$ for the
Gaussian P-spin glass model under the assumption RS and 1-RSB, showing that we
find the same results in AdrianoGauss .

Starting from the RS assumption, if we set $P=2$ in the quenched pressure of
the Gaussian P-spin glass model (128), we get

$\small\begin{array}[]{lll}\mathcal{A}_{RS}(\beta^{{}^{\prime}},\lambda)&=&\dfrac{\beta^{\prime}\,{}^{2}\bar{q}}{2\left(1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}\right)}-\dfrac{1}{2}\ln{\left[1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}\right]}+\dfrac{\beta^{\prime}\,{}^{2}}{4}\bar{q}^{2}\,.\end{array}$
(188)

Extremizing the statistical pressure in (188) w.r.t. the order parameter
$\bar{q}$ we find the following

######  Corollary 3.

The minimum of the quenched statistical pressure in (188) is achieved for

$\begin{array}[]{lll}\bar{q}=0&\mathrm{if}&\beta^{{}^{\prime}}\leq
1-\lambda,\\\ \\\
\bar{q}=\dfrac{\beta^{{}^{\prime}}-(1-\lambda)}{\beta^{\prime}\,{}^{2}}&\mathrm{if}&\beta^{{}^{\prime}}>1-\lambda.\end{array}$
(189)

###### Proof.

By setting equal to zero the derivative with respect to $\bar{q}$ of the eq.
(188), we get

$\small\bar{q}=\dfrac{\beta^{\prime}\,{}^{2}\bar{q}}{1-\lambda-\beta^{\prime}\,{}^{2}\bar{q}}$
(190)

this equation give us two possible solution of the order parameter, namely

$\small\begin{array}[]{lll}\bar{q}=0&\mathrm{and}&\bar{q}=\dfrac{\beta^{{}^{\prime}}-(1-\lambda)}{\beta^{\prime}\,{}^{2}}\end{array}$
(191)

Now, computing the second derivative of the quenched pressure and replacing
the values of $\bar{q}$ found in (191), we get

$\small\begin{array}[]{lll}\dfrac{\partial^{2}\mathcal{A}(\beta^{{}^{\prime}},\lambda)}{\partial\bar{q}^{2}}\Big{|}_{\bar{q}=0}=\dfrac{\beta^{\prime}\,{}^{2}}{2\left(1-\lambda\right)^{2}}\left[\left(1-\lambda\right)^{2}-\beta^{\prime}\,{}^{2}\right],\\\
\\\
\dfrac{\partial^{2}\mathcal{A}(\beta^{{}^{\prime}},\lambda)}{\partial\bar{q}^{2}}\Big{|}_{\bar{q}=(\beta^{{}^{\prime}}-1+\lambda)/\beta^{\prime}\,{}^{2}}=\beta^{{}^{\prime}}(\beta^{{}^{\prime}}-(1-\lambda)).\end{array}$
(192)

Thus, from the study of the sign of the previous equations we get the proof. ∎

Moving on the 1RSB case, if we set $P=2$ in (144), we find

$\small\begin{array}[]{lll}A_{1RSB}(\beta^{{}^{\prime}},\lambda)=&\dfrac{\beta^{\prime}\,{}^{2}}{4}\bar{q}_{2}^{2}-\dfrac{\beta^{\prime}\,{}^{2}}{4}\theta\left(\bar{q}_{2}^{2}-\bar{q}_{1}^{2}\right)-\dfrac{1}{2}\log\left[1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}\right]\\\
\\\
&+\dfrac{1}{2\theta}\log\left[\dfrac{1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}}{1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}-\beta^{\prime}\,{}^{2}\theta\left(\bar{q}_{2}-\bar{q}_{1}\right)}\right]\\\
\\\
&+\dfrac{1}{2}\dfrac{\beta^{\prime}\,{}^{2}\bar{q}_{1}}{1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}-\beta^{\prime}\,{}^{2}\theta\left(\bar{q}_{2}-\bar{q}_{1}\right)}\end{array}$
(193)

Following the same steps presented for the RS assumption, we set to zero the
derivatives of (193) respect to the order parameters $\bar{q}_{1}$ and
$\bar{q}_{2}$

$\small\begin{array}[]{lll}\bar{q}_{1}&=&\dfrac{\beta^{\prime}\,{}^{2}\bar{q}_{1}}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}-\beta^{\prime}\,{}^{2}\theta(\bar{q}_{2}-\bar{q}_{1})\right)^{2}}\,,\\\
\\\
\bar{q}_{2}-\bar{q}_{1}&=&\dfrac{\beta^{\prime}\,{}^{2}\left(\bar{q}_{2}-\bar{q}_{1}\right)}{\left(1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}-\beta^{\prime}\,{}^{2}\theta\left(\bar{q}_{2}-\bar{q}_{1}\right)\right)\left(1-\lambda+\beta^{\prime}\,{}^{2}\bar{q}_{2}\right)}\,.\end{array}$
(194)

It is immediate to verify that the previous system of equations admits only
the solution $\bar{q}_{1}=\bar{q}_{2}=\bar{q}$, with $\bar{q}$ of the form
presented in (189). Therefore the solution in 1RSB approssimation coincides
with the one in RS approximation. As a matter of fact, the other solution of
(194) for $\bar{q}_{1}\neq\bar{q}_{2}$, reads as

$\begin{array}[]{lll}(\bar{q}_{1};\bar{q}_{2})&=&\left(0\,;\dfrac{(\theta-2)(\lambda-1)-\sqrt{\theta^{2}(\lambda-1)^{2}-4\beta^{2}(\theta-1)}}{2\beta^{2}(\theta-1)}\right)\,,\end{array}$
(195)

however, it can be shown that this solution is neither a minimum nor a maximum
of (193), but a saddle point.

Thus, we can conclude this section with the following theorem

######  Theorem 6.

For the pairwise Gaussian spin glass model defined by the Hamiltonian

$H_{N}(\bm{z}|\bm{J})\coloneqq-\sqrt{\dfrac{1}{2N}}\displaystyle\sum\limits_{i,j=1}^{N,N}\,J_{ij}z_{i}z_{j}\;\;\;\mathrm{where}\;\;\;J_{ij}\sim\mathcal{N}(0,1)$
(196)

the RS solution is exact.

###### Proof.

The proof consists in showing that the 1RSB bound for the free energy gives
the same result of the RS approximation. ∎

##  3 Conclusions and outlooks

In this paper we have generalized two rigorous mathematical methods for two
different models, namely P-spin glass model and Gaussian P-spin model with
$P>2$. We face up to RS and 1RSB assumptions, comparing an approach closer to
Mathematical Physics, namely transport equation, and another linked to
Statistics and Probability, namely Guerra’s interpolating technique. We reach
the same results via both techniques, proving that they are both valid for
these models and we recovered the same expression in gardner ; GrossMezard ;
Tala4 ; Tala5  for P-spin glass and crisanti ; Tala3  for Gaussian P-spin
glass models.

Having the possibility to investigate the problem with mathematical tools from
different fields could be powerful and could give us non-trivial result.

In addiction we showed that, as far as Gaussian spin glass model concerns, the
replica symmetry expression for quenched statistical pressure is exact for
$P=2$ case, in contrast to $P>2$ case.

Further researches should be adressed to a deeper physical interpretation of
this results, in particular on replica symmetry breaking assumptions. Another
lines could be the rigorous mathematically confirmation of results in KRSB
assumption, with K finite in GrossMezard  and the approach of more challenging
models, such as dense associative networksdense .

##  Appendix A Proof of (24)-(25)

In this Appendix we show the computation of expression which is used either in
Guerra’s interpolating scheme or transport equations. Let’s start with (24).
Using the notation $\Delta X=X-\bar{X}$ and exploiting Newton’s binomial, we
get

$\begin{array}[]{lll}\langle
m^{P}\rangle&=&\Big{\langle}(m-\bar{m}+\bar{m})^{P}\Big{\rangle}\\\ \\\
&=&\displaystyle\sum\limits_{k=0}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{m}^{P-k}\Big{\langle}(m-\bar{m})^{k}\Big{\rangle}\\\ \\\
&=&(1-P)\bar{m}^{P}+P\bar{m}^{P-1}\langle
m\rangle+\displaystyle\sum\limits_{k=2}^{P}\begin{pmatrix}P\\\
k\end{pmatrix}\bar{m}^{P-k}\Big{\langle}(m-\bar{m})^{k}\Big{\rangle}\,.\end{array}$
(197)

In the same way (using $q_{12}$ instead of $m$ and $\bar{q}$ instead of
$\bar{m}$) we prove (25).

##  Appendix B Proof of Lemma 2

We prove the $t$ derivative of statistical pressure in 1RSB assumption.

$\displaystyle
d_{t}\mathcal{A}=\frac{1}{N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\mathcal{W}_{2}\omega\left[\dfrac{\beta^{{}^{\prime}}J_{0}N}{2}m^{P}(\bm{\sigma})-N\psi\,m(\bm{\sigma})\right.$
$\displaystyle\left.+\frac{\beta^{{}^{\prime}}J}{2\sqrt{t}}\sqrt{\dfrac{1}{2N^{P-1}}}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{i_{1}\cdots,i_{{}_{P}}}\sigma_{i_{1}}\cdots\sigma_{i_{P}}-\frac{1}{2\sqrt{1-t}}\sum_{a=1}^{2}\left(A_{a}\displaystyle\sum\limits_{i=1}^{N}J_{i}^{(a)}\sigma_{i}\right)\right]=$
$\displaystyle=-\psi\langle m\rangle+\frac{\beta^{{}^{\prime}}J_{0}}{2}\langle
m^{P}\rangle+B_{1}+B_{2}+B_{3},$ (198) $\displaystyle B_{1}$
$\displaystyle=\sqrt{\frac{1}{2N^{P-1}t}}\frac{\beta^{{}^{\prime}}J}{2N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\mathcal{W}_{2}\omega\left(\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}z_{\bm{i}}\sigma_{i_{1}}\cdots\sigma_{i_{P}}\right)=$
$\displaystyle=\sqrt{\frac{1}{2N^{P-1}t}}\frac{\beta^{{}^{\prime}}J}{2N}\displaystyle\sum\limits_{i_{1},\cdots,i_{{}_{P}}=1}^{N,\cdots,N}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\partial_{z_{\bm{i}}}\left(\mathcal{W}_{2}\omega\left(\sigma_{i_{1}}\cdots\sigma_{i_{P}}\right)\right)=\frac{\beta^{\prime}\,{}^{2}J^{2}}{4}\left[1+(\theta-1)\langle
q_{12}^{P}\rangle_{2}-\theta\langle q_{12}^{P}\rangle_{1}\right],$ (199)
$\displaystyle B_{2}$
$\displaystyle=-\frac{A_{1}}{2N\sqrt{1-t}}\sum_{i}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\mathcal{W}_{2}\omega\left(z_{i}^{(1)}\sigma_{i}\right)=-\frac{A_{1}}{2N\sqrt{1-t}}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\partial_{z_{i}^{(1)}}\mathcal{W}_{2}\omega\left(\sigma_{i}\right)$
$\displaystyle=\frac{A_{1}^{2}}{2}\left[1+(\theta-1)\langle
q_{12}\rangle_{2}-\theta\langle q_{12}\rangle_{1}\right],$ (200)
$\displaystyle B_{3}$
$\displaystyle=-\frac{A_{2}}{2N\sqrt{1-t}}\sum_{i}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\mathcal{W}_{2}\omega\left(z_{i}^{(2)}\sigma_{i}\right)=-\frac{A_{2}}{2N\sqrt{1-t}}\mathbb{E}_{0}\mathbb{E}_{1}\mathbb{E}_{2}\partial_{z_{i}^{(2)}}\mathcal{W}_{2}\omega\left(\sigma_{i}\right)$
$\displaystyle=\frac{A_{2}^{2}}{2}\left[1+(\theta-1)\langle
q_{12}\rangle_{2}\right].$ (201)

Rearranging together we obtain the thesis.

## Acknowledgments

The authors are grateful to Adriano Barra for several fruitful discussions.
LA acknowledge Unisalento and INFN for partial financial support.
LA acknowledge partial financial support by the grant “BULBUL” (bando di
collaborazione industriale scientifica e tecnologica tra Italia ed Israele).

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