# Path Integral Approach Unveils the Role of Complex Energy Landscape for
Activated Dynamics of Glassy Systems

Tommaso Rizzo  ISC-CNR, UOS Rome, Università “Sapienza”, Piazzale A. Moro 2,
I-00185, Rome, Italy  Dip. Fisica, Università “Sapienza”, Piazzale A. Moro 2,
I–00185, Rome, Italy

###### Abstract

The complex dynamics of an increasing number of systems is attributed to the
emergence of a rugged energy landscape with an exponential number of
metastable states. To develop this picture into a predictive dynamical theory
I discuss how to compute the exponentially small probability of a jump from
one metastable state to another. This is expressed as a path integral that can
be evaluated by saddle-point methods in mean-field models, leading to a
boundary value problem. The resulting dynamical equations are solved
numerically by means of a Newton-Krylov algorithm in the paradigmatic
spherical $p$-spin glass model that is invoked in diverse contexts from
supercooled liquids to machine-learning algorithms. I discuss the solutions in
the asymptotic regime of large times and the physical implications on the
nature of the ergodicity-restoring processes.

##  I Introduction

The emergence of a rugged free energy landscape with many minima and saddles
is a paradigm often invoked to explain complex dynamical phenomena ranging
from supercooled liquids Charbonneau et al. (2014) to the performance of
widely used algorithm in machine-learning and inference Ros et al. (2019a);
Mannelli et al. (2020). Powerful methods developed originally in the spin-
glass literature Mézard et al. (1987) allow to obtain a rather complete
description of the landscape in a growing number of statistical physics model.
On the debit side the resulting picture is essentially limited to static
properties of the landscape, like the energy of the metastable states, their
free energy and notably their number, the so-called configurational entropy.
Thus understanding the exact way in which the landscape shapes the dynamics is
largely an open problem. The question is particularly urgent in the context of
supercooled liquids where the rugged landscape paradigm is at the core of the
Random-First-Order-Transition (RFOT) theory Kirkpatrick et al. (1989); Wolynes
and Lubchenko (2012).

More than thirty years after its formulation RFOT is still one of the major
competing theories in the ongoing debate on the nature of the Glass transition
Rizzo (2020a). In a nutshell the theory posits that the physics of supercooled
liquids is the same of Spin-Glass (SG) models displaying one-step of Parisi’s
Replica-Symmetry-Breaking (1RSB) Mézard et al. (1987). The mean-field versions
of these models display an ergodicity-breaking transition at a dynamical
temperature $T_{d}$ where the phase space splits into many metastable states
that trap the dynamics; at lower temperatures, the configurational entropy,
i.e. the log of the number of metastable states, decreases eventually
vanishing at a static temperature $T_{s}$. Ergodicity breaking between $T_{d}$
and $T_{s}$ is a mean-field artifact and one expects that in real systems
ergodicity is restored through droplet-like excitations. Furthermore the size
of these excitations must diverge as the configurational entropy vanishes
leading eventually to a genuine ergodicity-breaking transition at $T_{s}$.
RFOT originated from the realization Kirkpatrick and Thirumalai (1987) that
similar features had been discussed in various unrelated (and themselves
controversial) theories of supercooled liquids, most notably: 1) dynamics at
the ergodicity-breaking transition is the same of the (avoided) Mode-Coupling-
Theory (MCT) of supercooled liquids Götze (2008), 2) within the Adams-Gibbs-Di
Marzio theory Gibbs and DiMarzio (1958); Adam and Gibbs (1965), the glass
transition is driven by a correlation length that diverges at the Kauzmann
temperature where the configurational entropy vanishes.

Efforts to validate the theory have been driving theoretical, numerical and
experimental research for years. At the theoretical level RFOT has been
substantiated by a number of results arguing that mean-field models of
supercooled liquids exhibit 1RSB Mézard and Parisi (2012); Monasson (1995)
including the solution of the limit of infinite physical dimensions
Charbonneau et al. (2014, 2017). Numerically, the MCT phenomenology is well
documented Kob and Andersen (1995); Kob (1999) as well as the increase of
dynamic Kob et al. (1997); Flenner et al. (2014) and static correlation
lengths Berthier et al. (2005); Biroli et al. (2008). Experimentally, the
observation of a decreasing configurational entropy dates back to the 40’s
while more recently RFOT has inspired measurements of non-linear
susceptibilities Albert et al. (2016). Furthermore the analogy with
supercooled liquids has also led to the discovery that off-equilibrium
relaxational dynamics of mean-field Spin-glass models display aging
Cugliandolo and Kurchan (1993) and it is an active line of research Folena et
al. (2020); Altieri et al. (2020).

In spite of this huge body of work, consensus on the validity of the theory is
still lacking. One of the problem is that RFOT-inspired literature often
focuses on quantities, namely point-to-set correlation lengths and
configurational entropy, whose actual relevance for the problem of the glass
transition, that instead is dynamical in its essence, can also be questioned.
Besides, predictions are often merely qualitatively, which is a problem given
that e.g. the observed static length increases are too modest to convince the
community that they actually drive the slowing down of the dynamics.

Figure 1: Supercooled liquids display caging and hopping in real space (left)
that correspond to evolution in the rugged phase space of mean-field models
(right).

To make progress it would be important to obtain RFOT predictions that are
both quantitative and dynamical. The main challenge is that current
theoretical knowledge is mostly limited to mean-field models that display
ergodicity-breaking at $T_{d}$: in order to recover ergodicity between $T_{d}$
and $T_{s}$ and obtain realistic predictions we have to go beyond mean-field,
intense efforts in this line of research are currently underway Baity-Jesi et
al. (2018a, b); Carbone et al. (2020); Hartarsky et al. (2019); Ros et al.
(2019b); Stariolo and Cugliandolo (2019); Stariolo and Cugliandolo (2020); Ros
et al. (2021). In recent years progress has been made for the temperature
region close to the dynamical temperature $T_{d}$ Rizzo (2014); Rizzo and
Voigtmann (2015); Rizzo (2016a). It is now possible to describe qualitatively
and quantitatively how the ergodicity-breaking MCT transition is turned into a
dynamical crossover in mean-field SG Rizzo (2016b) (due to finite-size
effects) and most importantly in some finite-dimensional models Rizzo and
Voigtmann (2020). In this paper I consider instead the region between $T_{d}$
and $T_{s}$ where metastable states are present and discuss how to compute the
transition rate, that is the exponentially small probability of a jump from an
equilibrium state to another occurring in a finite time. To make contact with
the phenomenology of supercooled liquids we have to remember that below the
experimental MCT transition temperature a particle is trapped most of the time
into a cage formed by the surrounding particles and diffusion occurs through
hopping i.e. sudden rare jumps from a cage to another (see fig. 1). Mean-field
models capture caging through the appearance of metastable states and the
study of jumps in the free energy landscape initiated in this paper is
essential to a quantitative description of hopping in real space.

###  I.1 Main Results

The main results to be discussed in the paper are: i) a path integral method
to compute the transition rate ii) a Newton-Krylov algorithm yielding the
numerical solution of the corresponding dynamical equations iii) an asymptotic
analysis of the solutions in the regime of large times and iv) some non-
trivial implications on the ergodicity-restoring processes.

Figure 2: left to right, top to bottom: density plot of the six functions
$C(t,t^{\prime})$, $C^{dt}(t,t^{\prime})$, $\mathcal{T}\hat{R}(t,t^{\prime})$,
$\mathcal{T}\hat{R}^{dt}(t,t^{\prime})$,
$\mathcal{T}^{2}\hat{X}(t,t^{\prime})$,
$\mathcal{T}^{2}\hat{X}^{dt}(t,t^{\prime})$ for $-\mathcal{T}\leq
t,t^{\prime}\leq\mathcal{T}$ and $\mathcal{T}=40$ for the Spherical 3-SG model
at $T=1/1.695<T_{d}$, (numerical solution with $\Delta t=\mathcal{T}/2000$).
Too large values close the diagonal are not shown for clarity (white).

At the methodological level the problem is formulated as a path integral over
Langevin dynamic trajectories that can be computed though saddle-point methods
in mean-field models. The problem displays some important differences with
respect to the standard relaxational dynamics Sompolinsky and Zippelius
(1982); Crisanti et al. (1993); Cugliandolo and Kurchan (1993), namely the use
of replicas and the need to explicitly handle the divergent path integral. I
have focused on the paradigmatic spherical $p$-spin SG model Crisanti and
Sommers (1992); Crisanti et al. (1993) but the method is fairly general and
the equations can be derived with some effort for other mean-field systems
e.g. supercooled liquids in large dimension Kurchan et al. (2016); Maimbourg
et al. (2016); Manacorda et al. (2020). Another more important difference
follows from to the fact that while in ordinary relaxation dynamics one only
fixes the initial condition, in order to study activated dynamics one has to
fix both the initial and final conditions. This difference manifests itself at
the level of the dynamical equations: while relaxational equations display
first-order time derivatives the equations obtained here are second-order. One
must then solve a more difficult boundary value problem instead of a simpler
initial value problem 111This is also the main difference with the earlier
method of Lopatin and Ioffe (1999, 2000) and more recently of Ros et al.
(2021) that allow to study some special activated processes through
relaxational dynamics equations but cannot be used to study transition rates
between generic equilibrium states.. Indeed while the relaxational dynamical
equations can be solved at times $t+\Delta t$ iteratively using the solution
at times $t^{\prime}\leq t$ the activated dynamical equations must be solved
simultaneously at all times, something for which no standard algorithm exists.
A successful solution strategy has been developed based on three elements: 1)
Newton’s method on discretized equations, 2) Krylov methods with physical
preconditioning to invert the Jacobian, 3) Richardson extrapolation to reach
the continuum limit. The whole technology can be again exported to other
problems, with possible computational complexity issues due to the specific
dynamical order parameter.

The paper is organized as follows. In the remaining of this section I will
give a compact presentation of the main results, leaving the details for the
body of the paper and the appendices. In particular in section I.1.1 the
object I compute is introduced and its general properties discussed. Later on
I discuss the asymptotic behavior of the solutions that is interesting for a
number of physical and technical reasons. From the physical point of view the
most interesting outcome of the computation is associated to the ergodicity
restoring processes as discussed in sec. I.1.3. The actual dynamical equations
are given later in section (II), their solutions will be studied in various
regimes (free, ergodic and activated). The numerical solution is challenging
and will be discussed in sec. II.6. In section III I will give some concluding
perspectives.

####  I.1.1 The Transition Rate

Figure 3: Spherical 3-SG at $T=1/1.695<T_{d}$. Left: instantaneous energy of
the trajectory for for $\mathcal{T}=8,\,16,\,24,\,32,\,40$, each curve was
obtained from finite $N_{t}$ solutions through Richardson extrapolation to
$\Delta t=0$ (see text). Center: Same data with times rescaled with
$\mathcal{T}$. The dotted line is the $\mathcal{T}=\infty$ limit obtained by
linear extrapolation in $1/\mathcal{T}$. Right: The energy at $t=0$ plotted
vs. $1/\mathcal{T}$. Convergence to to the $\mathcal{T}=\infty$ limit with
$1/\mathcal{T}$ corrections is clearly visible, the continuous lines are
linear fits on the last two and three points. The energy varies continuously
along the trajectory from the equilibrium value $e_{eq}=-\beta/2=-0.8475$ at
the initial and final time to larger values at intermediate times. Note that
the (extrapolated) maximum value $e(0)=-0.8106$ at $t=0$ is larger than the
threshold energy $e_{th}=-0.8311$, as computed from eqs. (12,13) in
Cugliandolo and Kurchan (1993) .

The object considered is the transition rate $T_{2\mathcal{T}}(\sigma|\tau)$
defined as the probability that the system is in configuration $\sigma$ at
time $t_{fin}=\mathcal{T}$ given that it was in configuration $\tau$ at time
$t_{in}=-\mathcal{T}$. It is convenient to actually consider the following
object that, due to detailed balance, is symmetric with respect to the
exchange of $\sigma$ and $\tau$:

$\hat{T}_{2\mathcal{T}}(\sigma,\tau)\equiv
T_{2\mathcal{T}}(\sigma|\tau)e^{{\beta\over 2}H(\sigma)-{\beta\over
2}H(\tau)}\ .$ (1)

An integral representation of Langevin dynamics is used and $\sigma$,$\tau$
are chosen as generic equilibrium configurations. At low temperatures
$\hat{T}(\sigma,\tau)$ is exponentially small in the system size $N$ in mean-
field models, therefore an annealed average
$P_{eq}(\sigma)P_{eq}(\tau)\hat{T}(\sigma,\tau)$ would interfere with the
equilibrium measure of $\sigma$ and $\tau$ and the correct procedure is to
consider the quenched average

$[\ln\hat{T}]\equiv\sum_{\sigma,\tau}P_{eq}(\sigma)P_{eq}(\tau)\,\ln\hat{T}(\sigma,\tau)\
\ .$ (2)

One can resort to the replica method to eliminate the logarithm, besides, if
quenched disorder is present, the corresponding averages (represented by an
overline in the following) require the introduction of additional replicas of
the initial and final configurations.

One can argue that the rate is self-averaging, meaning that most couples
$(\sigma,\tau)$ display the same rate 222This can be shown computing $O(n)$
corrections to the rate where $n$ is the replica number., thus, given an
initial configuration $\sigma$, the total number of configurations with rate
equal to the typical value given by the average
$\overline{[\ln\hat{T}_{2\mathcal{T}}]}$ is equal to the total number of
equilibrium configurations $\tau$, i.e. the exponential of the entropy,
$e^{S}$. Neglecting exponentially small corrections, the total probability of
jumping to one of the equilibrium configurations is thus $e^{[\ln\hat{T}]+S}$
and must be smaller than one leading to the bound:

$\overline{[\ln\hat{T}_{2\mathcal{T}}]}+S\leq 0\ .$ (3)

Now, after a finite time a system in equilibrium will be in another
equilibrium configuration correlated with the initial one, therefore the
probability to be in a generic equilibrium configuration (that is uncorrelated
to the initial one) must be smaller than one meaning that at any finite
$\mathcal{T}$ the above bound should not be saturated. This also implies that
even for $T>T_{d}$ if $\mathcal{T}<\infty$ one must consider the logarithm of
the rate (and thus resort to the replica method with replica number $n=0$) in
order not to interfere with the equilibrium measure on the initial and final
conditions. On the other hand ergodicity implies that when $\mathcal{T}$ goes
to infinity the probability measure will be flat over the $e^{S}$ equilibrium
configurations and the average rate should become equal to $e^{-S}$ saturating
the bound.

At the mean-field level the expression for the logarithm of the rate can be
computed by the saddle-point approximation meaning that the thermodynamic
limit $N\rightarrow\infty$ is always taken before the
$\mathcal{T}\rightarrow\infty$ limit. Thus one can compute the quantity

$\mathcal{A}\equiv\lim_{\mathcal{T}\rightarrow\infty}\lim_{N\rightarrow\infty}{1\over
N}\left(\overline{[\ln\hat{T}_{2\mathcal{T}}]}+S\right)\ .$ (4)

In 1RSB models, as discussed in the introduction, the dynamical temperature
$T_{d}$ marks the onset of activated dynamics and this results in the two
limits ceasing to commute:

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle 0\,\ \
\mathrm{for}\ T\geq T_{d}$ (5) $\displaystyle\mathcal{A}$ $\displaystyle<$
$\displaystyle 0\,\ \ \mathrm{for}\ T<T_{d}$ (6)

In the following sections the expression for the average transition rate of
the spherical $p$-spin model for $T>T_{s}$ is given. The Hamiltonian of the
model is given by:

$H=\sum_{p=1}^{\infty}\sum_{i_{1}<\dots<i_{p}}J_{i_{1}\dots
i_{p}}s_{i_{1}}\dots s_{i_{p}}$ (7)

where the $N$ spins verify the global spherical constraint $\sum_{i}s_{i}=N$,
the $J$’s are quenched Gaussian random variables of zero mean and variance:

$\overline{J_{i_{1}\dots i_{p}}^{2}}={\mu_{p}\,p!\over 2N^{p-1}}\,.$ (8)

The following function is defined for convenience:

$f(x)\equiv\sum_{p=1}^{\infty}\mu_{p}\,x^{p}\ .$ (9)

The expression for the rate $\overline{[\ln\hat{T}_{2\mathcal{T}}]}$ depends
on six real functions $C(t,t^{\prime})$, $\hat{R}(t,t^{\prime})$,
$\hat{X}(t,t^{\prime})$, $C^{dt}(t,t^{\prime})$, $\hat{R}^{dt}(t,t^{\prime})$
and $\hat{X}^{dt}(t,t^{\prime})$ defined for $-\mathcal{T}\leq
t,t^{\prime}\leq\mathcal{T}$. Two additional functions $\mu(t)$ and
$\hat{\mu}(t)$ enforce the spherical constraint leading to $C(t,t)=1$ and
$\hat{R}(t,t)=1/2$ for all $t$. Extremization of the expression leads to eight
non-linear integro-differential equations (albeit two are redundant due to
symmetries) that will be discussed in the next section. The physical meaning
of $C(t,t^{\prime})$ and $C^{dt}(t,t^{\prime})$ is straightforward:

$\displaystyle C(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle s_{i}(t)s_{i}(t^{\prime})\rangle]}$ (10)
$\displaystyle C^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle s_{i}(t)\rangle\langle
s_{i}(t^{\prime})\rangle]}$ (11)

where the square brackets mean averages with respect to the dynamical
trajectories at fixed initial and final configurations $(\sigma,\tau)$ and
fixed disorder. Thus $C(t,t^{\prime})$ is the average correlation between
configurations visited by the same trajectory at times $t$ and $t^{\prime}$
while $C^{dt}(t,t^{\prime})$ is the correlation between configurations visited
by different trajectories (hence the suffix dt). It follows that the equations
must satisfy the boundary conditions
$C(\pm\mathcal{T},\pm\mathcal{T})=C^{dt}(\pm\mathcal{T},\pm\mathcal{T})=1$ and
$C(\pm\mathcal{T},\mp\mathcal{T})=C^{dt}(\pm\mathcal{T},\mp\mathcal{T})=0$.

In section II.4 the $T>T_{d}$ regime will be discussed, here one can show that
in the $\mathcal{T}\rightarrow\infty$ limit the saddle-point equations admit a
solution in which the six functions are expressed in terms of the equilibrium
correlation $C_{eq}(0,t)$ and that the rate tends to $-S$ leading to
$\mathcal{A}=0$. This is possible because $C_{eq}(0,\infty)=0$ and the
boundary condition $C(\pm\mathcal{T},\mp\mathcal{T})=0$ can be satisfied for
$\mathcal{T}\rightarrow\infty$ by the equilibrium solution. This is no longer
true for $T<T_{d}$ because equilibrium dynamics is trapped and
$C_{eq}(0,\infty)\neq 0$, as a consequence for $T<T_{d}$ the average logarithm
of the rate is smaller than $-S$ also in $\mathcal{T}\rightarrow\infty$ limit
meaning that $\mathcal{A}$ becomes negative continuously at $T_{d}$. In fig. 2
the numerical solution for the spherical SG model with $p=3$ is shown for
$T=1/1.695<T_{d}$ and $\mathcal{T}=40$. The transition from the ergodic
$(T>T_{d})$ to the activated ($T<T_{d}$) regime is marked by a qualitative
change in the solutions. As it will be shown in section II.4 in the ergodic
phase at large $\mathcal{T}$ the various functions approach equilibrium time-
translational-invariant forms (see eqs. (76 ,77, 78)) and density plots like
those of fig. (2) tend to become symmetric with respect to the $t=t^{\prime}$
axes. Instead in the activated phase the density plots for $C(t,t^{\prime})$
and $C^{dt}(t,t^{\prime})$ display the block structure visible in fig. (2) at
all values of $\mathcal{T}$, corresponding to the fact that the system close
to the initial and final time performs essentially a relaxational dynamics in
the corresponding states and jumps from one state to the other for
$t,t^{\prime}\approx 0$. A quantity that also displays a qualitative change is
the instantaneous intensive energy $e(t)$ of the system along the trajectory
from one equilibrium configuration to another one. By definition we have that
$e(\pm\mathcal{T})$ is equal to the equilibrium value $e_{eq}$ at the
corresponding temperature. In general $e(t)$ is larger than $e_{eq}$ at
intermediate times but in the limit $\mathcal{T}\rightarrow\infty$ it tends to
be equal to $e_{eq}$ at all times for $T>T_{d}$, while for $T<T_{d}$
$e(t)>e_{eq}$ also in the large $\mathcal{T}$ limit and indeed one finds that
$e(t)\approx e_{u}(t/\mathcal{T})$ where $e_{u}(x)$ is a universal function
independent of $\mathcal{T}$ as shown by figure 3. This kind of asymptotic
behavior is also shared by the solution as we will further discuss in the
following.

Let us now mentions for completeness a few technical features of the problem
that will be further discussed in the following sections. At all temperatures,
even in the ergodic phase $T>T_{d}$, the solutions cannot be expressed in
terms of the equilibrium correlation if $\mathcal{T}$ is finite, however the
equations can be solved analytically at finite $\mathcal{T}$ in the free case
($T=\infty$) in which the system performs a Brownian motion on the $N-1$
dimensional sphere (see section II.3). In general the expression for the
average rate requires the computation of a path integral with an infinite
normalization factor, a well-known pathology that is typically discussed in
the context of the harmonic oscillator Zinn-Justin (2002); Parisi (1988). In
the non-interacting case one can use the harmonic oscillator formulas to
derive the expression of the rate as a function of $\mathcal{T}$. Knowledge of
the rate in the non-interacting case provides an alternative way to compute
the rate at finite temperature without having to deal with the infinite
normalization factor. Differentiating the saddle-point expression of the rate
with respect to the inverse temperature $\beta$ one gets indeed a finite
expression that yields the correct derivative when evaluated on the solutions
of the saddle-point equations. The rate at finite $\mathcal{T}$ and $\beta$
can then be obtained by integration in $\beta$ starting from the exact
$\beta=0$ result.

####  I.1.2 Asymptotic Behavior

The equations in the activated phase $T<T_{d}$ can only be studied numerically
for finite $\mathcal{T}$. One then faces the problem of extrapolating the
results to $\mathcal{T}\rightarrow\infty$ in order to avoid transient effects.

Figure 4: Spherical 3-SG at $T=1/1.695<T_{d}$. Top, left: $C(-\mathcal{T},t)$
vs. $t+\mathcal{T}$ for $\mathcal{T}=8,\,16,\,24,\,32,\,40$, the dashed line
is the equilibrium dynamics that is trapped inside a state and has a plateau
at $0.6340$; right: same vs. $t/\mathcal{T}$. Bottom, left: $C(0,t)$ vs. $t$
for $\mathcal{T}=8,\,16,\,24,\,32,\,40$, right: same vs. $t/\mathcal{T}$. Each
curve was obtained from finite $N_{t}$ solutions through Richardson
extrapolation to $\Delta t=0$ (see text).

The values of $\mathcal{T}$ that it was possible to reach numerically allow to
identify clearly the asymptotic behavior for $\mathcal{T}\rightarrow\infty$ in
the activated region. In particular in section (II.4) we will show see that
for finite time differences $|t-t^{\prime}|=O(1)\ll\mathcal{T}$ the functions
$C(t,t^{\prime})$, $\hat{R}(t,t^{\prime})$ and $\hat{X}(t,t^{\prime})$ verify
the equilibrium relationships corresponding to fluctuation-dissipation theorem
and time-translational invariance meaning that on finite time-scales the
trajectories are essentially equilibrium trajectories in a self-induced
slowly-varying field. Instead on the large time scales
$|t-t^{\prime}|=O(\mathcal{T})$ the solutions approach universal functions
independent of $\mathcal{T}$:

$\displaystyle C(t,t^{\prime})$ $\displaystyle\approx$ $\displaystyle
C_{u}(t/\mathcal{T},t^{\prime}/\mathcal{T})$ (12) $\displaystyle
C^{dt}(t,t^{\prime})$ $\displaystyle\approx$ $\displaystyle
C_{u}^{dt}(t/\mathcal{T},t^{\prime}/\mathcal{T})$ (13)
$\displaystyle\hat{R}(t,t^{\prime})$ $\displaystyle\approx$
$\displaystyle\mathcal{T}^{-1}\hat{R}_{u}(t/\mathcal{T},t^{\prime}/\mathcal{T})$
(14) $\displaystyle\hat{R}^{dt}(t,t^{\prime})$ $\displaystyle\approx$
$\displaystyle\mathcal{T}^{-1}\hat{R}_{u}^{dt}(t/\mathcal{T},t^{\prime}/\mathcal{T})$
(15) $\displaystyle\hat{X}(t,t^{\prime})$ $\displaystyle\approx$
$\displaystyle\mathcal{T}^{-2}\hat{X}_{u}(t/\mathcal{T},t^{\prime}/\mathcal{T})$
(16) $\displaystyle\hat{X}^{dt}(t,t^{\prime})$ $\displaystyle\approx$
$\displaystyle\mathcal{T}^{-2}\hat{X}_{u}^{dt}(t/\mathcal{T},t^{\prime}/\mathcal{T})$
(17) $\displaystyle\mu(t)$ $\displaystyle\approx$
$\displaystyle\mu_{u}(t/\mathcal{T})$ (18) $\displaystyle\hat{\mu}(t)$
$\displaystyle\approx$
$\displaystyle\mathcal{T}^{-1}\hat{\mu}_{u}(t/\mathcal{T})$ (19)

The above behavior is clearly visible in fig. (4) for the function
$C(t,t^{\prime})$. This also implies that the density plots for
$C(t,t^{\prime})$ and $C^{dt}(t,t^{\prime})$ at different (large) value of
$\mathcal{T}$ are indistinguishable. Note also that $\hat{R}(t,t^{\prime})$
and $\hat{X}(t,t^{\prime})$ are small for $|t-t^{\prime}|=O(\mathcal{T})$
while they are finite close to the diagonal, see fig. 2.

Plugging the above asymptotic expressions in the full dynamical equations one
sees that the universal functions solve the dynamical equations with the
second derivatives dropped, similarly to what happens for equilibrium dynamics
at $T_{d}$ and for off-equilibrium dynamics Crisanti et al. (1993);
Cugliandolo and Kurchan (1993) . Closing those equations would allow to work
directly at $\mathcal{T}\rightarrow\infty$ and is an open problem that is left
for future work. At any rate inspection of the equation suggests that the
solution should approach their $\mathcal{T}=\infty$ limit with $1/\mathcal{T}$
corrections. This is indeed supported by the data as can be seen considering
e.g. the instantaneous (intensive) energy $e(t)$ along a trajectory, see the
right panel of fig. (3).

One should note that the asymptotic structure of the solutions is utterly
different from that of metastability in ferromagnetism. The corresponding
computation describes the transition rate from the metastable minimum to the
stable one and leads to an instantonic equation in which the second order
derivatives is not dropped in the asymptotic limit. As a consequence even if
$\mathcal{T}\rightarrow\infty$ the jump effectively occurs in time window
centered around $t=0$ that remains finite and does not scale with
$\mathcal{T}$. Another more striking difference with metastability in
ferromagnetism occurs when we consider the ergodicity-restoring processes as
discussed next.

####  I.1.3 Ergodicity Restoring Processes

This work is focused on the exponentially small probability that the system
jumps to another equilibrium state in a finite time, which is complementary to
the problem of determining the exponentially large time-scale $\tau_{erg}$
over which the system finds itself into another equilibrium state with finite
probability.

Given that, according to the discussion before eq. (3) the exponentially small
probability to jump to an equilibrium state with typical rate is $p=e^{[\ln
T]+S}$ it is natural to expect that such a probability becomes finite on a
time-scale of order $1/p$. This is indeed what happens for ferromagnetism,
e.g. in the Curie-Weiss model at low temperatures where two metastable states
are present. The following discussion shows instead that the connection does
not hold in presence of an exponential number of metastable states, i.e. a
finite configurational entropy. In order to see this it is convenient to
introduce the average of the rate at inverse temperature $\beta$ over initial
and final configurations that are in equilibrium at different inverse
temperatures $\beta_{in}$ and $\beta_{fin}$:

$[\ln
T]_{\beta_{in},\beta,\beta_{fin}}\equiv\sum_{\sigma\,,\tau}P_{eq}^{(\beta_{fin})}(\sigma)P_{eq}^{(\beta_{in})}(\tau)\ln
T^{(\beta)}_{2\,\mathcal{T}}(\sigma|\tau)\ ,$ (20)

where $P_{eq}^{(\beta)}(\sigma)$ is the Boltzmann distribution at inverse
temperature $\beta$ and $T^{(\beta)}_{2\,\mathcal{T}}(\sigma|\tau)$ is the
transition rate from configuration $\tau$ to $\sigma$ due to Langevin dynamics
at inverse temperature $\beta$. The quantity that generalizes ${\cal A}$ as
given in eq. (4) is then:

${\cal
A}(\beta^{\prime}|\beta)\equiv\lim_{\mathcal{T}\rightarrow\infty}\lim_{N\rightarrow\infty}{1\over
N}\left(\overline{[\ln
T_{2\mathcal{T}}]}_{\beta,\beta,\beta^{\prime}}+S(\beta^{\prime})\right)\ ,$
(21)

An expression of the above quantity can be easily obtained in terms of second-
order dynamical equations as a straightforward generalization of the case
where both the initial and final configurations are at equilibrium, see
appendix E. One can thus compute the total probability of jumping from an
equilibrium state to a state with a different energy $E^{\prime}\neq E_{eq}$,
as

$p(E^{\prime})\propto\exp N{\cal A}(\beta^{\prime}|\beta)$ (22)

where naturally $\beta^{\prime}$ is such that $E^{\prime}$ is the equilibrium
energy at the (inverse) temperature $\beta^{\prime}$. An explicit computation
(see appendix E) shows that

$\displaystyle\left.\frac{dp(E^{\prime})}{dE^{\prime}}\right|_{E^{\prime}=E_{eq}}$
$\displaystyle=$ $\displaystyle 0\,\ \ \mathrm{for}\ T\geq T_{d}$
$\displaystyle\left.\frac{dp(E^{\prime})}{dE^{\prime}}\right|_{E^{\prime}=E_{eq}}$
$\displaystyle>$ $\displaystyle 0\,\ \ \mathrm{for}\ T<T_{d}$

The behavior for $T>T_{d}$ implies that the probability $p(E^{\prime})$ has a
maximum for $E^{\prime}=E_{eq}$, this was to be expected due to the fact that
$p(E_{eq})=1$ (see eq. (5)) saturates the bound $p(E^{\prime})\leq 1$. Instead
for $T<T_{d}$ we have $p(E^{\prime})\gg p$ for $E^{\prime}\geq E_{eq}$ and the
system has a finite probability to jump to one of these higher energy states
on a time scale $1/p(E^{\prime})$ exponentially smaller than $1/p(E_{eq})$:

$1/p(E^{\prime})\ll 1/p(E_{eq})\,\ \ \mathrm{for}\ T<T_{d},E^{\prime}>E_{eq}$
(23)

On the other hand the transition rate from a configuration with energy
$E^{\prime}$ back to a configuration with energy $E$ obeys the detailed
balance condition:

$e^{-\beta H(\tau)}T_{\Delta t}(\tau^{\prime}|\tau)=e^{-\beta
H(\tau^{\prime})}T_{\Delta t}(\tau|\tau^{\prime})\ ,$ (24)

taking into account that there are $S$ configurations with energy $E$ and
$S^{\prime}$ configurations with energy $E^{\prime}$ we find that the
probability to jump from a state with energy $E^{\prime}$ back to an
equilibrium state is

$p(E^{\prime})e^{\Delta S-\beta\Delta E}\gg p(E^{\prime})\,,$ (25)

i.e. it is exponentially larger than $p(E^{\prime})$ given that, by
definition, the free energy $-S+\beta E$ has its minimum on the equilibrium
states. Therefore, after the system has jumped to a state with higher energy
$E^{\prime}>E_{eq}$ it will jump back to a generic equilibrium state on a
scale exponentially smaller than $1/p(E^{\prime})$, i.e. instantaneously on
that scale. This implies that an intermediate jump to one of the exponentially
many metastable states with $E^{\prime}>E_{eq}$ provides a more efficient path
for restoring ergodicity than a direct jump to another equilibrium state and
thus the ergodic scale is smaller than $1/p(E_{eq})$, at variance with
ferromagnetism where there are only two metastable states. It is to be
expected that $p(E^{\prime})$ reaches a maximum $p(E_{max})\ll 1$ at some
$E_{max}>E_{eq}$ for $T<T_{d}$. The true ergodic time $\tau_{erg}$ should be
identified with $1/p(E_{max})$ but could be even smaller, a detailed analysis
of $p(E^{\prime})$ and of the correct $\tau_{erg}$ is left for future work.

####  I.1.4 Reproducibility

The full commented code to solve numerically the equations is provided online
together with the required initialization files, see the ancillary files
section of Rizzo (2020b).

##  II Analysis of The Saddle-Point equations of the Spherical $p$-Spin-Glass
Model

###  II.1 The order parameter and its meaning

In the appendix I will derive the expression for the logarithm of the rate
$\mathcal{A}$ associated to Langevin dynamics of the Spherical $p$-SG model in
terms on an order parameter determined through saddle-point equations. The
order parameter is a couple of $2\times 2$ matrices of functions of two times
$t$ and $t^{\prime}$ on the square $-\mathcal{T}\leq
t,t^{\prime}\leq\mathcal{T}$:

$C\equiv\left(\begin{array}[]{cc}C(t,t^{\prime})&\hat{R}_{2}(t,t^{\prime})\\\
\hat{R}_{1}(t,t^{\prime})&\hat{X}(t,t^{\prime})\end{array}\right)\,$ (26)
$C^{dt}\equiv\left(\begin{array}[]{cc}C^{dt}(t,t^{\prime})&\hat{R}_{2}^{dt}(t,t^{\prime})\\\
\hat{R}_{1}^{dt}(t,t^{\prime})&\hat{X}^{dt}(t,t^{\prime})\end{array}\right)\,$
(27)

The physical meaning of $C(t,t^{\prime})$ and $C^{dt}(t,t^{\prime})$ has been
discussed already: $C(t,t^{\prime})$ is the correlation on the same trajectory
(therefore $C(t,t)=1$ at all times in the spherical and Ising model) while
$C^{dt}(t,t^{\prime})$ measures the correlations between the configurations
visited by different trajectories. Since by definition all trajectories have
the same initial and final condition we have

$C^{dt}(\pm\mathcal{T},t)=C(\pm\mathcal{T},t)\ .$ (28)

The correlations are obviously symmetric with respect to
$(t,t^{\prime})\rightarrow(t^{\prime},t)$:

$\displaystyle C(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
C(t^{\prime},t)$ (29) $\displaystyle C^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle C^{dt}(t^{\prime},t)\ .$ (30)

Furthermore given that the measure over the trajectories is invariant under
time reversal we have an additional symmetry with respect to the exchange of
the initial and final configuration. Given that $t_{in}=-t_{fin}$ this
symmetry translates into:

$\displaystyle C(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
C(-t,-t^{\prime})$ (31) $\displaystyle C^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle C^{dt}(-t,-t^{\prime})$ (32)

For the $\hat{R}$ components of the order parameter we have:

$\displaystyle\hat{R}_{1}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle\hat{x}_{i}(t)s_{i}(t^{\prime})\rangle]}$ (33)
$\displaystyle\hat{R}_{1}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle\hat{x}_{i}(t)\rangle\langle
s_{i}(t^{\prime})\rangle]}$ (34)

where $\hat{x}$ is an auxiliary variable of the dynamics (see the appendix).
They translate into:

$\displaystyle\hat{R}_{1}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{\left[{\delta\langle
s_{i}(t^{\prime})\rangle\over\beta\,\delta h(t)}\right]}-{1\over 2}{d\over
dt}C(t,t^{\prime})$ (35) $\displaystyle\hat{R}_{1}^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle\overline{\left[{\delta\ln
D(\sigma,\tau)\over\beta\,\delta h(t)}\langle
s_{i}(t^{\prime})\rangle\right]}-{1\over 2}{d\over dt}C^{dt}(t,t^{\prime})$
(36)

Thus $\hat{R}_{1}(t,t^{\prime})$ is connected to the response of the time
average of the spin over trajectories to a field $h(t)$. The functions
$\hat{R}_{2}(t,t^{\prime})$ and $\hat{R}_{2}^{dt}(t,t^{\prime})$ are equal to
the l.h.s.’s of the above equations with the exchange $t\leftrightarrow
t^{\prime}$. Thus while neither function is symmetric with respect to
$t\leftrightarrow t^{\prime}$ they are related through two functions
$\hat{R}(t,t^{\prime})$ and $\hat{R}^{dt}(t,t^{\prime})$ such that

$\hat{R}_{1}(t,t^{\prime})=\hat{R}_{2}(t^{\prime},t)=\hat{R}(t^{\prime},t)$
(37)
$\hat{R}_{1}^{dt}(t,t^{\prime})=\hat{R}_{2}^{dt}(t^{\prime},t)=\hat{R}^{dt}(t^{\prime},t)$
(38)

which implies that the matrices (26) and (27) are symmetric. On the other hand
time-reversal invariance implies that $\hat{R}_{1}(t,t^{\prime})$,
$\hat{R}_{2}(t,t^{\prime})$, $\hat{R}_{1}^{dt}(t,t^{\prime})$,
$\hat{R}_{2}^{dt}(t,t^{\prime})$, are symmetric with respect to
$(t,t^{\prime})\rightarrow(-t,-t^{\prime})$ (because $t_{in}=-t_{fin}$ as
above). For the $\hat{X}$ components we have:

$\displaystyle\hat{X}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle\hat{x}_{i}(t)\hat{x}_{i}(t^{\prime})\rangle]}$
(39) $\displaystyle\hat{X}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\overline{[\langle\hat{x}_{i}(t)\rangle\langle\hat{x}_{i}(t^{\prime})\rangle]}$
(40)

The physical meaning is also associated to particular responses that takes a
simple form in the ergodic phase, see section II.4. The above formulas imply
that both $\hat{X}(t,t^{\prime})$ and $\hat{X}^{dt}(t,t^{\prime})$ are
symmetric with respect to $t\leftrightarrow t^{\prime}$ and to time-reversal
$(t,t^{\prime})\rightarrow(-t,-t^{\prime})$.

###  II.2 The Saddle Point Equations

The saddle-point equations can be written in a compact form which is also
suitable for numerical integration considering the space of $2\times 2$
matrices whose components are functions of two times $t$ and $t^{\prime}$ on
the square $-\mathcal{T}\leq t,t^{\prime}\leq\mathcal{T}$. The generic element
of this space can be written as

$A\equiv\left(\begin{array}[]{cc}C_{A}(t,t^{\prime})&\hat{R}_{2,A}(t,t^{\prime})\\\
\hat{R}_{1,A}(t,t^{\prime})&\hat{X}_{A}(t,t^{\prime})\end{array}\right)\,$
(41)

Given two elements $A$ and $B$ in the above space we have a natural definition
of the product that generalizes the matrix product (it corresponds to exactly
to ordinary matrix products if times are discretized). For a real function
$B(x)$ we also define the element-wise function $B[A]$ such that

$\displaystyle C_{B[A]}(t,t^{\prime})$ $\displaystyle\equiv$ $\displaystyle
B(C_{A}(t,t^{\prime}))\,,$ $\displaystyle\hat{R}_{2,B[A]}(t,t^{\prime})$
$\displaystyle\equiv$ $\displaystyle
B^{\prime}(C_{A}(t,t^{\prime}))\,\hat{R}_{2,A}(t,t^{\prime})\,,$
$\displaystyle\hat{R}_{1,B[A]}(t,t^{\prime})$ $\displaystyle\equiv$
$\displaystyle
B^{\prime}(C_{A}(t,t^{\prime}))\,\hat{R}_{1,A}(t,t^{\prime})\,,$
$\displaystyle\hat{X}_{B[A]}(t,t^{\prime})$ $\displaystyle\equiv$
$\displaystyle B^{\prime}(C_{A}(t,t^{\prime}))\,\hat{X}_{A}(t,t^{\prime})+$
$\displaystyle+$ $\displaystyle
B^{\prime\prime}(C_{A}(t,t^{\prime}))\,\hat{R}_{1,A}(t,t^{\prime})\hat{R}_{2,A}(t,t^{\prime})\,.$

We define also:

$M\equiv\left(\begin{array}[]{cc}-{1\over
2\Gamma_{0}}\delta^{\prime\prime}(t,t^{\prime})+\hat{\mu}(t)\delta(t,t^{\prime})&\mu(t)\delta(t,t^{\prime})\\\
\mu(t)\delta(t,t^{\prime})&-{2\over\Gamma_{0}}\delta(t,t^{\prime})\end{array}\right)\,$
(42)

and

$T\equiv\left(\begin{array}[]{cc}0&\delta(t,t^{\prime})\\\
\delta(t,t^{\prime})&o\end{array}\right)\,\ .$

Note that due to the spherical constraint the operator $M$ depends on two
additional quantities $\mu(t)$ and $\hat{\mu}(t)$, see appendix A.3. In order
to write down the saddle-point equations it is useful to introduce two
additional objects $\Lambda$ and $\Lambda^{dt}$ that are also $2\times 2$
matrices of two-time functions. Another useful quantity is:

$\delta_{\mp}\equiv\left(\begin{array}[]{cc}\hat{R}_{\Lambda,1}(t,\mp\mathcal{T})C(\mp\mathcal{T},t^{\prime})&\hat{R}_{\Lambda,1}(t,\mp\mathcal{T})\,\hat{R}_{2}(\mp\mathcal{T},t^{\prime})\\\
C_{\Lambda}(t,\mp\mathcal{T})C(\mp\mathcal{T},t^{\prime})&C_{\Lambda}(t,\mp\mathcal{T})\hat{R}_{2}(\mp\mathcal{T},t^{\prime})\end{array}\right)\,\
.$ (43)

With the above definitions the saddle-point equations of the spherical model
derived in the appendix take the following compact expressions:

$\Lambda=-{\beta^{2}\over 2}f^{\prime}[C]\,,\ \Lambda^{dt}=-{\beta^{2}\over
2}f^{\prime}[C^{dt}]$ (44) $\displaystyle MC+\,T\,\Lambda\,T\,C$
$\displaystyle+$
$\displaystyle+(n-1)\,T\,\Lambda^{dt}\,T\,\,C^{dt}+\delta_{-}+\delta_{+}$
$\displaystyle=$ $\displaystyle I$ (45) $\displaystyle
MC^{dt}+\,T\,\Lambda^{dt}\,T\,\,C+\,T\,\Lambda\,T\,C^{dt}$ $\displaystyle+$
$\displaystyle+(n-2)\,T\,\Lambda^{dt}\,T\,C^{dt}+\delta_{-}+\delta_{+}$
$\displaystyle=$ $\displaystyle 0$ (46)

The quantities $\mu(t)$ and $\hat{\mu}(t)$ are unknown and must be determined
self-consistently imposing the conditions:

$C(t,t)=1\,,\ \ \hat{R}(t,t)={1\over 2}\ \ \ \forall t\ .$ (47)

Due to the presence of the operator $M$ that contains second-order derivatives
the equations must be also supplemented with the following boundary
conditions:

$\displaystyle C(\pm\mathcal{T},t^{\prime})$ $\displaystyle=$ $\displaystyle
C(t^{\prime},\pm\mathcal{T})$ (48)
$\displaystyle\hat{R}_{2}(\pm\mathcal{T},t^{\prime})$ $\displaystyle=$
$\displaystyle\hat{R}_{1}(t^{\prime},\pm\mathcal{T})$ (49) $\displaystyle
C^{dt}(\pm\mathcal{T},t^{\prime})$ $\displaystyle=$ $\displaystyle
C^{dt}(t^{\prime},\pm\mathcal{T})$ (50)
$\displaystyle\hat{R}_{2}^{dt}(\pm\mathcal{T},t^{\prime})$ $\displaystyle=$
$\displaystyle\hat{R}^{dt}_{1}(t^{\prime},\pm\mathcal{T})$ (51)

Additional boundary conditions follow from the properties of the initial and
final configurations:

$C(\pm\mathcal{T},\pm\mathcal{T})=1\,,\ \ C(\pm\mathcal{T},\mp\mathcal{T})=0$
$C^{dt}(\pm\mathcal{T},\pm\mathcal{T})=1\,,\ \
C^{dt}(\pm\mathcal{T},\mp\mathcal{T})=0$

The above boundary conditions are sufficient to compute the r.h.s. of the
saddle point equations (45,46) for a generic $C$ and $C^{dt}$ but given the
structure of the equation and the physical meaning of the order parameter the
actual solution verifies the additional symmetries discussed previously. In
particular the symmetry of the problem under the exchange between the initial
and final configuration implies:

$\mu(t)=\mu(-t)\,,\ \ \hat{\mu}(t)=\hat{\mu}(-t)\ .$ (52)

In the numerical analysis times are discretized, thus $C$ and $C^{dt}$ become
actual matrices and the above symmetries allow a four-fold reduction of the
memory required to store them.

###  II.3 The free case

In the infinite temperature limit the interaction vanishes and the system
performs a free Brownian motion on the $N-1$ dimensional sphere. In this case
one can find the analytic solution of the saddle-point equations which, in
turn, is useful to guess an initial solution to feed to the Newton’s algorithm
at finite temperature. Besides, as we discussed in the introduction, it allows
to bypass the non-renormalizability of the expression for
$\overline{[\ln\hat{T}]}$ by integrating with respect to the temperature. The
saddle point equations (45,46) become in the free case:

$\displaystyle MC=I\,,\ \ MC^{dt}=0$ (53)

The first equation corresponds to:

$\displaystyle-2\hat{R}_{1}(t,t^{\prime})+\mu(t)C(t,t^{\prime})$
$\displaystyle=0$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}C(t,t^{\prime})+\mu(t)\hat{R}_{1}(t,t^{\prime})+\hat{\mu}(t)C(t,t^{\prime})$
$\displaystyle=$ $\displaystyle\delta(t-t^{\prime})$
$\displaystyle-2\hat{X}(t,t^{\prime})+\mu(t)\hat{R}_{2}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle\delta(t-t^{\prime})$ $\displaystyle-{1\over
2}{d^{2}\over
dt^{2}}\hat{R}_{2}(t,t^{\prime})+\mu(t)\hat{X}(t,t^{\prime})+\hat{\mu}(t)\hat{R}_{2}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle 0$

Note that the equations for $C$ do not depend on $C^{dt}$. To solve them one
can start noticing that the conditions (47) plugged into the first equation
lead to

$\mu(t)=1\ \ \ \forall t\,.$ (54)

Then the first equation and the symmetries
$\hat{R}_{2}(t,t^{\prime})=\hat{R}_{1}(t^{\prime},t)$,
$C(t,t^{\prime})=C(t^{\prime},t)$ and the third equation lead to

$\displaystyle\hat{R}_{1}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 2}C(t,t^{\prime})$ (55)
$\displaystyle\hat{R}_{2}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 2}C(t,t^{\prime})$ (56)
$\displaystyle\hat{X}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle-{1\over
2}\delta(t-t^{\prime})+{1\over 4}C(t,t^{\prime})\ .$ (57)

As a consequence the correlation $C(t,t^{\prime})$ is determined by the
following equation:

$-{1\over 2}{d^{2}\over dt^{2}}C(t,t^{\prime})+\left({1\over
2}+\hat{\mu}(t)\right)C(t,t^{\prime})=\delta(t-t^{\prime})$

To be solved with boundary conditions
$C(\pm\mathcal{T},t)=C(t,\pm\mathcal{T})$ and
$C(\pm\mathcal{T},\pm\mathcal{T})=1$, $C(\pm\mathcal{T},\mp\mathcal{T})=0$.
The equation has the solution

$C(t,t^{\prime})=C(|t-t^{\prime}|)\,,\ \ \hat{\mu}(t)=\hat{\mu}\,.$ (58)

To determine the function $C(x)$ it is convenient to define a quantity
$-\infty<a<1$ implicitly through the equation:

$\mathcal{T}={1\over 2\sqrt{a}}\,{\rm arctanh}\sqrt{a}\ .$ (59)

Figure 5: Left: The function $C(t,t^{\prime})$ in the free case for $a=0.999$
corresponding to $\mathcal{T}=2.07442$ (see text), Right: the function
$C^{dt}(t,t^{\prime})$.

We then have, see fig. (5):

$\displaystyle\hat{\mu}$ $\displaystyle=$ $\displaystyle{a-1\over 2}$ (60)
$\displaystyle C(x)$ $\displaystyle=$
$\displaystyle\cosh\sqrt{a}x-{1\over\sqrt{a}}\sinh\sqrt{a}x\ .$ (61)

Note that we also have $C(0)=1$ and $\dot{C}(0)=-1$ and these are general
results that hold also at finite temperature being a consequence of the
presence of the delta functions and of the symmetry
$C(t,t^{\prime})=C(t^{\prime},t)$. Let us discuss the limits of large and
small $\mathcal{T}$. For large values of $\mathcal{T}$ we have:

$a\approx 1-4\,e^{-4\mathcal{T}}\,,\ \hat{\mu}\approx-2\,e^{-4\mathcal{T}}\ \
\mathrm{for}\ \ \mathcal{T}\rightarrow\infty\ .$ (62)

Note that in the large $\mathcal{T}$ limit $\hat{\mu}$ tends to zero and the
function $C(x)$ tends to the equilibrium solution $C_{eq}(x)=e^{-x}$. On the
other hand $a$ becomes negative for $\mathcal{T}<1/2$:

$a\approx 6\left(\mathcal{T}-{1\over 2}\right)\ \mathrm{for}\ \
\mathcal{T}\approx{1\over 2}$ (63)

therefore $a^{1/2}$ is imaginary and the hyperbolic functions in (61) become
ordinary trigonometric functions:

$\displaystyle C(x)$ $\displaystyle=$
$\displaystyle\cos\sqrt{|a|}x-{1\over\sqrt{|a|}}\sin\sqrt{|a|}x\ .$ (64)

In the small $\mathcal{T}$ limit $a$ tends to minus infinity as

$a\approx-{\pi^{2}\over 16\mathcal{T}^{2}}\,,\ \hat{\mu}\approx-{\pi^{2}\over
32\mathcal{T}^{2}}\ \ \mathrm{for}\ \ \mathcal{T}\rightarrow 0$ (65)

leading to:

$C(x)\approx\cos{\pi x\over 4\mathcal{T}}\ \mathrm{for}\ \
\mathcal{T}\rightarrow 0$ (66)

The above expression correctly vanishes at $x=2\mathcal{T}$ but does not
display the correct behavior $C(x)\approx 1-x$ at small $x$. Indeed it is only
valid for $x=O(\mathcal{T})$ and the correct linear behavior is recovered for
$x\ll\mathcal{T}$. We now turn to the equations for $C^{dt}$:

$\displaystyle-2\hat{R}_{1}(t,t^{\prime})+\mu(t)C^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}C^{dt}(t,t^{\prime})+\mu(t)\hat{R}_{1}^{dt}(t,t^{\prime})+\hat{\mu}(t)C^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle-2\hat{X}^{dt}(t,t^{\prime})+\mu(t)\hat{R}_{2}^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}\hat{R}_{2}^{dt}(t,t^{\prime})+\mu(t)\hat{X}^{dt}(t,t^{\prime})+\hat{\mu}(t)\hat{R}_{2}^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle 0$

The condition $\mu(t)=1$ derived earlier, the first equation, the symmetries
$\hat{R}_{2}^{dt}(t,t^{\prime})=\hat{R}_{1}^{dt}(t^{\prime},t)$,
$C^{dt}(t,t^{\prime})=C^{dt}(t^{\prime},t)$ and the third equation lead to:

$\displaystyle\hat{R}_{1}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 2}\,C^{dt}(t,t^{\prime})$ (67)
$\displaystyle\hat{R}_{2}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 2}\,C^{dt}(t,t^{\prime})$ (68)
$\displaystyle\hat{X}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 4}\,C^{dt}(t,t^{\prime})$ (69)

and $C^{dt}(t,t^{\prime})$ obeys the equation

$-{1\over 2}{d^{2}\over dt^{2}}C^{dt}(t,t^{\prime})+\left({1\over
2}+\hat{\mu}\right)C^{dt}(t,t^{\prime})=0$ (70)

To be solved with boundary conditions
$C^{dt}(\pm\mathcal{T},t)=C^{dt}(t,\pm\mathcal{T})$ and
$C^{dt}(\pm\mathcal{T},\pm\mathcal{T})=1$,
$C^{dt}(\pm\mathcal{T},\mp\mathcal{T})=0$. The solution reads:

$\displaystyle C^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{\sqrt{1-a}\over a}\cosh(\sqrt{a}(t+t^{\prime}))+$
$\displaystyle-$ $\displaystyle{1-a\over a}\cosh(\sqrt{a}(t-t^{\prime}))\,,\
\mathrm{for}\ \mathcal{T}>1/2$ $\displaystyle C^{dt}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle{\sqrt{1-a}\over
a}\cos(\sqrt{|a|}(t+t^{\prime}))+$ $\displaystyle-$ $\displaystyle{1-a\over
a}\cos(\sqrt{|a|}(t-t^{\prime}))\,\ \mathrm{for}\ 0<\mathcal{T}<1/2$

In the limit $\mathcal{T}\rightarrow\infty$ we have $C^{dt}(t,t^{\prime})=0$
for any finite $t,t^{\prime}$. Close to the initial and finite time we have
instead

$C^{dt}(\pm\mathcal{T}+\delta t,\pm\mathcal{T}+\delta
t^{\prime})=C_{eq}(|\delta t+\delta t^{\prime}|)\ \mathrm{for}\ \
\mathcal{T}\rightarrow\infty.$

In the limit $\mathcal{T}\rightarrow 0$ we obtain instead:

$C^{dt}(t,t^{\prime})\approx\cos{\pi(t-t^{\prime})\over 4\mathcal{T}}\
\mathrm{for}\ \ \mathcal{T}\rightarrow 0$

Note that in this limit we have also $C^{dt}(t,t^{\prime})\approx
C(t,t^{\prime})$ meaning that all trajectories tend to follow the same path.

The computation of the transition rate requires to treat carefully the
divergences of the path integral and is given in the appendix. The final
result is:

${1\over N}\overline{[\ln\hat{T}]}=-{1\over 2}\ln 2\pi-{1\over
2}+(1+\hat{\mu})\mathcal{T}+{1\over 4}\ln{-\hat{\mu}\over 2}\ .$

From the above expression we see that the logarithm of the rate tends to minus
infinity as $\mathcal{T}$ goes to zero:

${1\over N}\overline{[\ln\hat{T}]}\approx-{\pi^{2}\over 32\mathcal{T}}\,,\ \ \
\mathcal{T}\approx 0\ .$

For $\mathcal{T}$ going to infinity one can see that although
$\hat{\mu}\approx-2\,e^{-4\mathcal{T}}$ the $O(\mathcal{T})$ divergences in
the last two terms cancel and the rate has a finite limit. This limit is
exactly equal to minus the entropy $S(\beta)$ of the spherical model at
infinite temperature ($\beta=0$), i.e. the logarithm of the surface of the
$N$-dimensional sphere of radius $\sqrt{N}$:

$\lim_{\mathcal{T}\rightarrow\infty}{1\over N}\overline{[\ln\hat{T}]}=-{1\over
2}\ln 2\pi-{1\over 2}=-S(0)\ .$

This is expected in the $\mathcal{T}\rightarrow\infty$ limit at any finite $N$
and implies that the two limits commute. One can show that the approach to the
$\mathcal{T}\rightarrow\infty$ limit is exponential:

${1\over N}\overline{[\ln\hat{T}]}\approx-S(0)-{1\over
2}e^{-4\,\mathcal{T}}\,,\ \ \mathcal{T}\gg 1\ .$

###  II.4 The Ergodic Phase

As explained in the introduction the ergodic phase defined by $T>T_{d}$ is
characterized by the fact that the $\mathcal{T}\rightarrow\infty$ limit
commutes with the $N\rightarrow\infty$ limit. In order to discuss the
solutions in this regime it is convenient to analyze first the ergodic limit
at finite $N$. While the present formalism is fully invariant under time
reversal to discuss this limit it is convenient to reintroduce the arrow of
time. In the ergodic limit $\mathcal{T}\rightarrow\infty$ dynamics looses any
dependence on the initial configuration and the transition rate obeys

$\lim_{\mathcal{T}\rightarrow\infty}T_{2\mathcal{T}}(\sigma|\tau)={e^{-\beta
H(\sigma)}\over Z}$ (71)

that, according to the previous definitions leads to:

$\lim_{\mathcal{T}\rightarrow\infty}\hat{T}_{2\mathcal{T}}(\sigma,\tau)={1\over
Z}e^{-{\beta\over 2}H(\sigma)-{\beta\over 2}H(\tau)}$ (72)

and the following exact result:

$\lim_{\mathcal{T}\rightarrow\infty}[\ln\hat{T}_{2\mathcal{T}}(\sigma,\tau)]=-\beta
E-\ln Z=-S$ (73)

In the ergodic limit we expect that trajectories close to the initial
condition at time $-\mathcal{T}$ are not influenced by the fact that we are
fixing the final configuration at time $\mathcal{T}$. This implies that they
are typical trajectories and since the initial configuration is weighted with
the equilibrium weight we expect that correlations and response are those
valid at equilibrium and in particular satisfy time-translational-invariance
(TTI) and fluctuation-dissipation theorem (FDT). To see the implications on
the functions it is convenient to start from the expressions of the six
functions as averages of $s_{i}(t)$ and $\hat{x}_{i}(t)$ obtained in
subsection II.1. It is then convenient to transform back to the variable
$\hat{s}$ introduced in the first steps of the integral represenation (see
appendix A.2) by writing

$\hat{x}_{i}=\hat{s}_{i}(t)-\dot{s}_{i}(t)\ .$ (74)

It is well known the equilibrium averages of $\hat{s}_{i}(t)$ are associated
to responses to a field at time $t$ and since according to (71) the
distribution at $\sigma$ is completely independent of what happens at any
finite time $t\ll\mathcal{T}$ we have for these times

$\langle\hat{s}(t)\rangle=0\,,\ \
\langle\hat{s}(t)\hat{s}(t^{\prime})\rangle=0\ .$ (75)

The above relationships plus FDT allow to derive the following expressions in
the ergodic limit:

$\displaystyle C(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
C_{eq}(|t-t^{\prime}|)$ (76) $\displaystyle\hat{R}_{2}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle\hat{R}_{1}(t^{\prime},t)={\rm
sign}(t-t^{\prime})\,{1\over 2}\,{d\over dt^{\prime}}C(t,t^{\prime})$ (77)
$\displaystyle\hat{X}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle-{1\over
4}{d^{2}\over dt\,dt^{\prime}}C(t,t^{\prime})$ (78)

Note that $\hat{R}(t,t^{\prime})$ is symmetric (which is not true in the
activated phase) and it is equal to $1/2$ on the diagonal according to the
saddle-point equations. The expression for $\hat{X}(t,t^{\prime})$ is valid
also for $t=t^{\prime}$ and thus it may be integrated, this is consistent with
the fact that $\hat{X}(t,t^{\prime})$ has a term proportional
$-\delta(t-t^{\prime})/2$ on the diagonal. The above properties have been
derived for $t,t^{\prime}\ll\mathcal{T}$, in order to study the region of
times close to $\mathcal{T}$ it is better to reverse the arrow of time,
following the same arguments we obtain the above equations with
$(t,t^{\prime})\rightarrow(-t,-t^{\prime})$ and since the functions are
symmetric with respect to this transformation we conclude that they are valid
at all times.

At large values of $\mathcal{T}$ the correlation between different
trajectories are different from zero only for times $t,t^{\prime}$ that are
both close to either $-\mathcal{T}$ or $\mathcal{T}$. To determine them we can
use the following relationship :

$\sum_{\tau}{e^{-\beta H(\tau)}\over Z}T_{\Delta
t}(\tau^{\prime}|\tau)T_{\Delta
t^{\prime}}(\tau^{\prime\prime}|\tau)={e^{-\beta H(\tau^{\prime})}\over
Z}T_{\Delta t+\Delta t^{\prime}}(\tau^{\prime\prime}|\tau^{\prime})$ (79)

that follows from the detailed balance condition:

$e^{-\beta H(\tau)}T_{\Delta t}(\tau^{\prime}|\tau)=e^{-\beta
H(\tau^{\prime})}T_{\Delta t}(\tau|\tau^{\prime})$ (80)

and from the general property:

$\sum_{\tau}T_{\Delta t^{\prime}}(\tau^{\prime\prime}|\tau)\,T_{\Delta
t}(\tau|\tau^{\prime})=T_{\Delta t+\Delta
t^{\prime}}(\tau^{\prime\prime}|\tau^{\prime})\ .$ (81)

Furthermore responses between different trajectories vanish because of
$\langle\hat{s}(t)\rangle=0$. In particular if we write times as
$t=\mp\mathcal{T}\pm\Delta t$ ($\Delta t\geq 0$) we have:

$\displaystyle C^{dt}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
C_{eq}(\Delta t+\Delta t^{\prime})$ (82)
$\displaystyle\hat{R}_{2}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\hat{R}_{1}^{dt}(t^{\prime},t)=\mp{1\over 2}\,{d\over
dt^{\prime}}C^{dt}(t,t^{\prime})$ (83)
$\displaystyle\hat{X}^{dt}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle{1\over 4}{d^{2}\over dt\,dt^{\prime}}C^{dt}(t,t^{\prime})\ .$
(84)

Note that the expression for $\hat{R}_{2}^{dt}(t,t^{\prime})$ changes sign
depending on weather we are close to $\mp\mathcal{T}$, indeed the second case
($+\mathcal{T}$) is obtained from the transformation
$(t,t^{\prime})\rightarrow(-t,-t^{\prime})$ applied to the first case
($-\mathcal{T}$). A solution of the saddle-point equations with the above
structure can be found only if $C_{eq}(\infty)=0$, thus the ergodic solution
does not exist below $T_{d}$ where $C_{eq}(\infty)=q$ in the thermodynamic
limit and the condition $C(\pm\mathcal{T},\mp\mathcal{T})=0$ cannot be
fulfilled. The above expressions are valid for both Ising and spherical
models, in addition for the spherical model we have:

$\mu(t)=\mu_{eq}\,,\ \ \ \hat{\mu}(t)=0\,.$ (85)

Note that all components are parameterized by the equilibrium correlation
$C_{eq}(t)$, let us see how an equation for $C_{eq}(t)$ can be derived in the
present context. In the formulation of the problem discussed in the appendix
one works with objects $\bm{Q(ab)}$. In the ergodic phase one can additionally
introduce a special class of objects $\bm{A(ab)}$ whose components $A(ab)$ and
$A^{dt}(ab)$ obey formulas (76,77,78) and (82,83,84) with $C_{eq}(t)$ replaced
by some generic function $C_{A}(t)$. In the following this kind of objects
will be loosely referred as having an ‘equilibrium” structure. A simple
computation shows that the equilibrium structure is preserved by the
application of a function, i.e. $f(\bm{A(ab)})$ has also the equilibrium
structure with:

$f(\bm{A(ab)})\rightarrow C_{f[A]}(t)=f(C_{A}(t))$ (86)

Similarly, given another equilibrium object $\bm{B(ab)}$ parameterized by a
function $C_{B}(t)$, one can show through a tedious computation that the
product has also the equilibrium structure with:

$\int\bm{A(ac)B(cb)dc}\rightarrow C_{AB}(t)$
$C_{AB}(t)=C_{A}(0)C_{B}(t)-\int_{0}^{t}C_{A}(t-s){dC_{B}(s)\over ds}ds$

The above equation holds for $C_{A}(\infty)=C_{B}(\infty)=0$ which is granted
by the fact that we are working in zero field and $f^{\prime}(0)=0$ so that
the overlap between different equilibrium states is zero. The last two
equations applied to the saddle-point equation (136) lead immediately to the
equilibrium equation:

${\dot{C}}(t)=-\,C(t)-{\beta^{2}\over
2}\int_{0}^{t}f^{\prime}(C(t-s))\,\dot{C}(s)\,ds$ (87)

where we have used $\mu=1+{\beta^{2}\over 2}f^{\prime}(1)$ that follows from
the condition $C(0)=-\dot{C}(0)=1$. The above equations is usually written as
Crisanti and Sommers (1992); Crisanti et al. (1993); Castellani and Cavagna
(2005):

$\displaystyle\dot{C}(t)$ $\displaystyle=$
$\displaystyle-\,C(t)+{\beta^{2}\over 2}f^{\prime}(C(t))(1-C(t))+$ (88)
$\displaystyle-$ $\displaystyle{\beta^{2}\over
2}\int_{0}^{t}(f^{\prime}(C(t-s))-f^{\prime}(C(t)))\,{dC\over ds}\,ds$

Where $C(\infty)$ is the solution of the equation

$C={\beta^{2}\over 2}f^{\prime}(C)(1-C)\ .$ (89)

This equation admits the solution $C=0$ at all temperature but develops an
additional non-zero solution at $T_{d}$ specified by the condition:

$1={\beta_{d}^{2}\over 2}(f^{\prime\prime}(C_{d})(1-C_{d})-f^{\prime}(C_{d}))$
(90)

In the pure $p$-spin models $f(x)=x^{p}$ we have

$T_{d}=\sqrt{{p(p-2)^{p-2}\over 2(p-1)^{p-1}}}\,,\ \ C_{d}(\infty)={p-2\over
p-1}\ \ .$ (91)

To conclude the discussion of the ergodic phase we have to show that for
$T>T_{d}$ the $\mathcal{T}\rightarrow\infty$ and $N\rightarrow\infty$ limits
commute and we have

$\mathcal{A}=0\,\ \ \mathrm{for}\ \,T_{d}<T<\infty$ (92)

In the previous section we have shown that the relationship is satisfied for
$\beta=1/T=0$ and thus it suffices to show that its derivative with respect to
the $\beta$ is also zero. In appendix E we show that the derivative of the
rate with respect to $\beta$ is ${\beta\over 2}{d\over d\,n}\int d{\bf a}d{\bf
b}f({\bf Q(a,b)})$. The above expression takes a very simple form in the
ergodic phase, indeed using the above equilibrium formulas one can show that
the generic formula (191) for the integrals takes a very simple form:

$\displaystyle\int\bm{B(ab)dadb}$ $\displaystyle=$
$\displaystyle(m+m^{\prime})C_{B}(0)+n\left(4\int
R_{2,B}(-\mathcal{T},t)dt+\int\hat{X}_{B}(t,t^{\prime})\,dt\,dt^{\prime}+(n-1)\int\hat{X}_{B^{dt}}(t,t^{\prime})\,dt\,dt^{\prime}\right)=$
(93) $\displaystyle=$ $\displaystyle(m+m^{\prime}+n)C_{B}(0)$

that leads to:

${d\over
d\beta}\lim_{\mathcal{T}\rightarrow\infty}\lim_{N\rightarrow\infty}{1\over
N}\overline{[\ln\hat{T}]}={\beta\over 2}f(1)\,,$ (94)

to be compared with

${1\over N}{dS(\beta)\over d\beta}=-{\beta\over 2}f(1)\ .$ (95)

We thus see that eq. (92) is verified:

$\lim_{\mathcal{T}\rightarrow\infty}\lim_{N\rightarrow\infty}{1\over
N}\overline{[\ln\hat{T}]}=-S(\beta)=-S(0)+{\beta^{2}\over 4}f(1)\ .$ (96)

The above relationship is valid also for Ising systems.

###  II.5 The Activated Phase

In the activated phase $T_{s}<T<T_{d}$ the saddle-point equations have been
solved numerically for the classic pure $p$-spin with $f(x)=x^{3}$ where
$\beta_{d}=1.63299$ and $\beta_{s}=1.70633$ Crisanti and Sommers (1992);
Crisanti et al. (1993); Castellani and Cavagna (2005). Due to the significant
resources needed to solve the equations, as discussed in the next section,
most of the analysis has been done at a single inverse temperature
$\beta=1.695$ for values of $\mathcal{T}=8,16,24,32,40$.

Figure 6: Spherical $p$-Spin Glass for $\beta=1.695$ and $p=3$. Solutions for
$\mathcal{T}=8\,(blue)$, $16\,(yellow),24\,(green),32\,(red),40\,(purple)$ on
various strips of the $-\mathcal{T}\leq t,t^{\prime}\leq\mathcal{T}$ plane.
The vertical scale of the $\hat{R}(t,t^{\prime})$ and $\hat{X}(t,t^{\prime})$
functions does not allows to see the diagonal region $t-t^{\prime}=O(1)$.

In fig. (6) we clearly see the features anticipated in the introduction, in
particular: i) the overlap between the initial (final) configuration and the
$t=0$ intermediate configuration $C(\pm\mathcal{T},0)$ tends to a finite value
in the $\mathcal{T}\rightarrow\infty$ limit, ii) the functions
$\hat{R}(t,t^{\prime})$, $\hat{R}^{dt}(t,t^{\prime})$,
$\hat{X}(t,t^{\prime})$, $\hat{X}^{dt}(t,t^{\prime})$ decrease with increasing
$\mathcal{T}$ for $|t-t^{\prime}|=O(\mathcal{T})$.

Figure 7: Spherical $p$-Spin Glass for $\beta=1.695$ and $p=3$. Scaled
solutions for $\mathcal{T}=8\,(blue)$,
$16\,(yellow),24\,(green),32\,(red),40\,(purple)$ on various strips of the
$-\mathcal{T}\leq t,t^{\prime}\leq\mathcal{T}$ plane. Figure 8: Spherical
$p$-Spin Glass for $\beta=1.695$ and $p=3$. The functions
$\hat{R}(t,t^{\prime})$ and
$\delta\hat{X}(t,t^{\prime})=\hat{X}(t,t^{\prime})+\delta(t-t^{\prime})/2$ for
$\mathcal{T}=8,16,24,32,40$ on various slices of the $-\mathcal{T}\leq
t,t^{\prime}\leq\mathcal{T}$ plane. The vertical and horizontal ranges are
appropriate to visualize the diagonal region $t-t^{\prime}=O(1)$. On this
scale the functions for different $\mathcal{T}$’s are almost
indistinguishable.

In fig. (7) the same data are rescaled to fully demonstrate the asymptotic
limit $\mathcal{T}\rightarrow\infty$ discussed in sec. I.1.2. The same-
trajectory functions $\hat{R}(t,t^{\prime})$ and $\hat{X}(t,t^{\prime})$
deviate from this scaling in the region $|t-t^{\prime}|=O(1)$ that goes to
zero on the scale of the plots for $\mathcal{T}\rightarrow\infty$. In the
region $|t-t^{\prime}|=O(1)$ $\hat{R}(t,t^{\prime})$ and
$\hat{X}(t,t^{\prime})$ converge to a finite limit asymptotically as shown in
fig. (8). One can also check that in that region $C(t,t^{\prime})$,
$\hat{R}(t,t^{\prime})$ and $\hat{X}(t,t^{\prime})$ satisfy the equilibrium
relationships (77) and (78) consistently with the fact that the system is in
equilibrium on scales much smaller than $\mathcal{T}$ in a field slowly
varying on a $O(\mathcal{T})$ timescale.

Figure 9: Spherical $p$-Spin Glass for $\beta=1.695$ and $p=3$. The auxiliary
functions $\mu(t)$ and $\hat{\mu}(t)$ for
$\mathcal{T}=8\,(blue),16\,(yellow),24\,(green),32\,(red),40\,(purple)$. The
$\mathcal{T}=40$ value in the rescaled plot for $\hat{\mu}(t)$ has not been
plot because the extrapolation from $N_{t}=2000$ seems not accurate enough on
that scale.

The auxiliary functions $\mu(t)$ and $\hat{\mu}(t)$ are shown in fig. (9).
Note that the $\mathcal{T}^{-1}$ scaling of $\hat{\mu}(t)$ is consistent with
the fact that $C(t,t^{\prime})$, $\hat{R}(t,t^{\prime})$ and
$\hat{X}(t,t^{\prime})$ satisfy equilibrium relationship on the diagonal
according to eq. (85). However $\mu(t)$ changes with time implying equilibrium
on finite time-scales but not on the global scale $O(\mathcal{T})$.

Figure 10: Spherical $p$-Spin Glass with $p=3$. Derivative of the total
transition probability with respect to the inverse temperature for
$\mathcal{T}=24,32$ and $\beta=1.55$, $1.6$, $1.65$, $1.695$, the lines are
second-order interpolations.

In fig. (10) we plot the derivative of $\overline{\langle\ln\hat{T}\rangle}+S$
with respect to the inverse temperature computed according to expression
(181). According to the result of the previous section this quantity should go
to zero in the $\mathcal{T}\rightarrow\infty$ limit for
$\beta<\beta_{d}=1.6329$. On the other hand the asymptotic behavior sets in
when $\mathcal{T}$ is larger than the equilibrium relaxation time that
diverges as $|T-T_{d}|^{-\gamma}$ where $\gamma=1.765$ is a MCT exponent
Crisanti et al. (1993); Caltagirone et al. (2012); Ferrari et al. (2012). Thus
at any $\mathcal{T}$, no matter how large, there is always a range of
temperatures $\Delta T\propto\mathcal{T}^{-1/\gamma}$ such that for
$T-T_{d}<\Delta T$, $\overline{\langle\ln\hat{T}\rangle}+S$ is smaller than
zero. The effect decreases with increasing $\mathcal{T}$ as the figure shows
but is still significant at the values of $\mathcal{T}$ that we could study.
On the other hand the curves seems to have converged to $-.47$ at
$\beta=1.695$ (as data from $\mathcal{T}=40$ also suggest) and this result,
supplemented with the information that it must be zero asymptotically at
$\beta_{d}$, allows for a rough estimate of the integral leading to

$\mathcal{A}=-0.014\ \,\mathrm{for}\ \beta=1.695$ (97)

###  II.6 Numerical Solution

####  II.6.1 Newton-Krylov Methods

To solve the equations numerically time was discretized in steps $\Delta
t=\mathcal{T}/N_{t}$ for integer $N_{t}$ up to $N_{t}=2000$. The six functions
can then be jointly represented as two real matrices $C$ and $C^{dt}$ of size
$2(2N_{t}+1)$ so that the total number of variables is $O(32N_{t}^{2})$. Using
standard formulas for discrete integrals and second-order derivatives (see
below) one obtains expressions with an $O(\Delta t^{2})$ error.

It turned out that the equations can be solved by Newton’s method. To
initialize the algorithm an approximate solution not too far away from the
correct solution can be obtained from the analytic solutions in the free case.
One can start from $\beta=0$ at finite $\mathcal{T}$ and switch on the
temperature, then at fixed temperature, $\mathcal{T}$ can be changed changing
the discretization parameter $\Delta t$ at fixed $N_{t}$ by small amounts. At
fixed $\mathcal{T}$ and $\beta$, $\Delta t$ can be reduced extrapolating the
result of a coarser grid to a finer grid (larger $N_{t}$) and using it as a
starting point for Newton’s method at the new $N_{t}$. The main technical
problem is that every iteration of Newton’s method requires to invert the
Jacobian of the equations, a $(32N_{t}^{2})\times(32N_{t}^{2})$ matrix. Even
exploiting the symmetries of the problem, with current technology exact
inversion of the Jacobian becomes unfeasible for $N_{t}$ of the order $70-80$
limiting the values of $\mathcal{T}$ that can be studied. $N_{t}$ could
instead be increased up to $2000$ using an approximate method for the well-
studied problem Saad (2003) of solving a very large linear system $A\,x=b$.
Specifically I used the Generalized Minimal Residue (GMRES) algorithm Saad and
Schultz (1986); Saad (2003) that requires the computation of the Krylov
subspace of order $k$, defined by the vectors
$\\{b,A\,b,A^{2}\,b,A^{3}\,b,\dots,A^{k}b\\}$. The advantage of Krylov methods
is that one always work with vectors $v$ and has to perform matrix
multiplications $A\,v$ without the need to store the full
$(32N_{t}^{2})\times(32N_{t}^{2})$ matrix $A$. In GMRES one searches for an
approximate solution in the Krylov sub-space of order $k$: an orthonormal
basis is obtained using a numerically stable Gram-Schmidt orthogonalisation
called Arnoldi iteration and the solution is then found by least squares
minimization of the linear equations in this space. The advantage of the
method is that the error decreases systematically increasing $k$, the drawback
is that it requires to generate and store all the $k$ vectors of the basis and
$k$ cannot be too small to obtain accurate solutions. Two important
ingredients that are key to the efficiency of the procedure are
preconditioning and compression to be discussed later. I wrote a code using
Mathematica (retrievable in the ancillary files section of Rizzo (2020b))
being able to reach values of $N_{t}=2000$ with $k=400$ using up to 90$\%$
memory on a cluster with 256 Giga of RAM.

Figure 11: Spherical $p$-Spin Glass with $p=3$, numerical solution for
$\beta=1.695$, $\mathcal{T}=24$ and $N_{t}=600$, $800$, $1000$, $1200$,
$1400$, $1600$, $1800$. Left: numerical values of $\mu(0)$, the lines are two
polynomial interpolation of the form $c_{0}+c_{2}\,\Delta t^{2}+c_{3}\,\Delta
t^{3}+c_{4}\,\Delta t^{4}$ over the data for $\\{1200$,$1400$,$1600$,$1800\\}$
and $\\{1000$,$1200$,$1400$,$1600\\}$. The polynomial extrapolations give
respectively $c_{0}=5.0290$ and $c_{0}=5.0288$. Right: from top to bottom data
for increasing values of $N_{t}$ (individual points are not distinguishable at
the scale of the plot). At the bottom there are two (indistinguishable) lines
obtained from polynomial interpolations performed separately for each time
$t$, one of the form $c_{0}+c_{2}\,\Delta t^{2}+c_{3}\,\Delta
t^{3}+c_{4}\,\Delta t^{4}$ over the data for
$\\{1200$,$1400$,$1600$,$1800\\}$, and the other of the form
$c_{0}+c_{2}\,\Delta t^{2}+c_{3}\,\Delta t^{3}$ over the data for
$\\{600$,$800$,$1000\\}$. Figure 12: Spherical $p$-Spin Glass with $p=3$,
numerical solution for $\beta=1.695$, $\mathcal{T}=24$ and various $N_{t}$.
Left: numerical values of $\hat{\mu}(0)$ for $N_{t}=600$, $800$, $1000$,
$1200$, $1400$, $1600$, $1800$, the lines are two polynomial interpolation of
the form $c_{0}+c_{2}\,\Delta t^{2}$ over the data for $\\{1600$,$1800\\}$ and
$\\{1400$,$1600\\}$. Right: from top to bottom data for $N_{t}=1000$ (blue),
$1400$ (yellow), $1800$ (green) (individual points are not distinguishable at
the scale of the plot). At the bottom there are two (indistinguishable) lines
(red, purple) obtained from polynomial interpolations of the form
$c_{0}+c_{2}\,\Delta t^{2}$ performed separately for each time $t$, using
respectively $N_{t}=\\{1200$,$1400\\}$ and $N_{t}=\\{1600$,$1800\\}$.

####  II.6.2 Richardson Extrapolation

Once the numerical solutions at fixed $\beta$ and $\mathcal{T}$ are obtained
for various $N_{t}$, a polynomial (Richardson) extrapolation is essential to
reach the $\Delta t=0$ limit and remove a few pathologies of the finite
$\Delta t$ solutions. Let us discuss first the order of the algorithm. For the
second derivatives appearing in the equations I have used the formula

$f^{\prime\prime}(t)\approx{f(t+\Delta t)+f(t-\Delta t)-2f(t)\over\Delta t}$
(98)

where

$\Delta t=\mathcal{T}/N_{t}\,.$ (99)

The above formula has a $O(\Delta t^{2})$ error if $f(t)$ has continuous
derivatives up to the third order. The integrals have been written as

$\int_{-\mathcal{T}}^{+\mathcal{T}}f(t)dt\approx{\Delta t\over
2}(f(-\mathcal{T})+f(+\mathcal{T}))+\sum_{i=-N_{t}+1}^{N_{t}-1}f(i\,\Delta
t)\Delta t\ .$ (100)

The above trapezoidal rule also has a $O(\Delta t^{2})$ error for a continuous
function. As mentioned before, the full algorithm has a $O(\Delta t^{2})$
error, however this is not trivial because the functions $C(t,t^{\prime})$,
$\hat{R}(t,t^{\prime})$ and $\hat{X}(t,t^{\prime})$ have discontinuous odd
derivatives for $t=t^{\prime}$. While the discontinuity of the first
derivative is canceled by the delta function the discontinuity of the third
derivative leads to a $O(\Delta t)$ error in expression (98) on the diagonal
$t=t^{\prime}$. However on the diagonal the order of the equation is not
$O(1)$ but $O(1/\Delta t)$ due to the presence of the delta function in eq.
(167) and of $\hat{X}(t,t^{\prime})$ (that can be written as a delta function
on the diagonal plus a regular part) in eq. (168), thus even if the absolute
error on the diagonal is $O(\Delta t)$, the relative error is $O(\Delta
t^{2})$. In Fig. (11) it is demonstrated that the error is indeed $O(\Delta
t^{2})$ for $\mu(t)$ and it is shown how that the $\Delta t\rightarrow 0$
limit is safely reached by polynomial extrapolations. The figures discussed in
the previous sections were all obtained by means of polynomial extrapolations
on the largest $N_{t}$ set available (up to $N_{t}=2000$ for
$\mathcal{T}=32,40$) using in most cases a fourth order form
$c_{0}+c_{2}\,\Delta t^{2}+c_{3}\,\Delta t^{3}+c_{4}\,\Delta t^{4}$ where the
vanishing of the linear term $\Delta t$ was imposed. One should note that the
finite $N_{t}$ curves often display pathologies due to the discretisation that
tend to be less severe increasing $N_{t}$. For instance in the right of fig.
(11) we see that $\mu(t)$ at finite $N_{t}$ displays cusps close to
$t=\pm\mathcal{T}$ that are absent for $N_{t}\rightarrow\infty$. It is
impressing how a simple polynomial extrapolation over $N_{t}$ leads to the
disappearance of these spurious features and allows to obtain accurate
predictions with relatively small values of $N_{t}$ that are indistinguishable
from extrapolations obtained from considerably larger values of $N_{t}$. The
same features are also seen in fig. (12), in this case corrections higher that
$O(\Delta t^{2})$ are so small that too high-order interpolation functions
overfit that data and it is convenient to use the form $c_{0}+c_{2}\,\Delta
t^{2}$. Besides we see that, even if the finite $N_{t}$ results for
$\hat{\mu}(t)$ have pronounced spurious cusps close to $t\pm\mathcal{T}$ and
may even have the wrong sign, the extrapolations are cuspless and negative for
all $t$.

####  II.6.3 Preconditioning

Efficient Krylov methods often require preconditioning which amount to reduce
the span of the eivengalues of $A$. The equations in the form (45) and (46)
are ill-conditioned because the operator $M$ contains second-order derivatives
leading to an unbounded continuous spectrum. To overcome this problem I have
multiplied equations (45) and (46) times $M^{-1}$. The corresponding Jacobian
turns out to have a discrete and bounded spectrum as can be seen numerically
using the fact that the Arnoldi diagonalization allows to obtain an
approximate set of eigenvalues and eigenvectors. More details on the procedure
can be found in the commented Mathematica codes provided in the ancillary
files section of Rizzo (2020b).

####  II.6.4 Compression

To compute the equations and the matrix-times-vector products $J\,c$ involving
the Jacobian $J$ we have to store huge matrices and perform multiplications
and element-wise operations, for which many highly-optimized and parallelized
libraries exist. One can solve eq. (45) and (46) for generic $C$ and $C^{dt}$
and verify that as expected the algorithm converges to solutions with the
required symmetries discussed in sec. II.1 . However it is useful to work from
the start in the subspace of solutions with the required symmetries obtaining
a four-fold reductions of the memory required to store $C$ and $C^{dt}$ and
thus the generic element of the Krylov subspace. Besides it turns out that
this allows to consider a smaller Krylov subspace to obtain the same level of
accuracy. Explicit use of the symmetry however requires an efficient procedure
to quickly compress and decompress the large matrices $C$ and $C^{dt}$ and the
elements of the Krylov subspace. The choice depends on the specific linear
algebra programming tool used.

##  III Concluding Perspectives

In this work I have shown how the transition rate can be computed in mean-
field models. As stated in the introduction, the main motivation to perform
such a computation is to obtain dynamical quantitative predictions not
accessible through static methods. Besides the most interesting physical
outcome of the computation is to shed new light on the role of the complex
landscape for ergodicity restoring processes as discussed in section I.1.3:
the conclusion is that to visit different equilibrium states it is more
convenient to make intermediate jumps to off-equilibrium metastable states
rather than direct jumps from one equilibrium state to another.

The methods discussed here can be applied to other mean-field systems, in the
introduction we mentioned supercooled liquids models in high dimensions but
the Langevin equation can be also viewed as an algorithm in the wider family
of (stochastic) gradient-descent algorithms that are widely used in the
context of machine-learning and statistical inference. The analysis of the
algorithm performance in the context of high-dimensional inference has been
initiated in Mannelli et al. (2020) unveiling a glassy phase that limits the
efficiency of the algorithm. Those results where obtained in the thermodynamic
limit where the evolution of the algorithm can be associated to relaxational
dynamic equations similar to those of the spherical $p$-spin SG model Crisanti
et al. (1993); Cugliandolo and Kurchan (1993) recently reconsidered in Folena
et al. (2020). On the hand other realistic problems have finite size and thus
the algorithm can overcome those thresholds albeit with larger convergence
times that can be studied by the methods presented here switching from the
initial value dynamical equations studied in Mannelli et al. (2020) to
boundary value dynamical equations. Concerning the extension to supercooled
liquid in high dimensions, the equations can be derived and solved in
principle by the GMRES algorithm although the actual implementation is likely
to be considerably more complex than for the spherical model Manacorda et al.
(2020). In this respect it is worth noticing that for more complex problems
one can use a Jacobian-free method, in which the product $J\,v$ required by
Newton-Krylov methods is approximated by $E(c+\epsilon\,v)-E(c))/\epsilon$ for
small $\epsilon$ and thus only the numerical computation of the equations
$E(c)$ is required. Furthermore other algorithms exist that do not require to
store the whole Krylov Space, e.g. Biconjugated Gradient Saad (2003); in
general they are less safe and controlled than GMRES, but an efficient
algorithm requiring less memory would be welcomed. Note that GMRES allows to
have an approximation for the spectrum of the true Jacobian, which could be
useful information to devise alternative algorithms.

We have mentioned that in the asymptotic regime $\mathcal{T}\rightarrow\infty$
the solutions solve $\mathcal{T}$-independent scaling equations with the time
derivatives dropped. In the context of off-equilibrium dynamics the same thing
happens and allows to establish a remarkable connection with the equations of
the static/replica framework Cugliandolo and Kurchan (1993). This grants that
the phase diagram and many non-trivial off-equilibrium quantities can be
determined without solving explicitly the dynamical equations and it would be
interesting to determine if there is also some static potential from which the
results of the $\mathcal{T}\rightarrow\infty$ limit can be recovered without
solving the dynamics. A natural question is weather such a static potential is
the celebrated Franz-Parisi potential Franz and Parisi (1997) that develops a
secondary minimum at $T_{d}$. One should indeed remember that, while the
minimum is associated to equilibrium states and configurational complexity, it
is not known if the potential difference between the secondary minimum and the
maximum is in fact associated to any quantity computed by actual dynamical
methods as those considered here.

An interesting open question is how one should expect the solutions to look
like at $T_{d}$, in particular in the asymptotic form. Some guidance could be
offered by the fact that in the $T\rightarrow T_{d}$ limit the results should
match somehow those obtained in Rizzo (2014); Rizzo and Voigtmann (2015);
Rizzo (2016a, b); Rizzo and Voigtmann (2020) by considering finite-
size/finite-dimensional systems directly at $T=T_{d}$ (from a technical point
of view the two approaches differ by which one of the limits $T\rightarrow
T_{d}$ or $N\rightarrow\infty$ is taken first). This suggests for instance
that the important physics occurs close to the plateau value of the overlap.

It should be noted that the dynamical equations have been solved by making a
Replica-Symmetric ansatz on the $n$ dynamical replicas introduced to compute
the logarithm of the rate (see eq. (139) in appendix B) and one may ask if it
yields correct results or a RSB ansatz should be used. While a full answer
requires a complicated analysis of the stability of the saddle-point
equations, an instability could nonetheless manifests itself in some
inconsistent value of some physical quantity, much as the RS entropy of the
Sherrington-Kirkpatrick model becomes negative at low temperatures Mézard et
al. (1987). This is not the case as the numerical solution did not show any
visible inconsistency suggesting that the RS ansatz is correct in the range of
temperatures considered.

To conclude we recall that in section I.1.3 it was shown that the presence of
many metastable states implies that the exponentially small probability that
the system jumps to another equilibrium state in a finite time is not
trivially related to the exponentially large time-scale $\tau_{erg}$ over
which the system jumps to another equilibrium state with finite probability. A
question worth of further investigation is weather the latter could also be
computed by similar methods.

###### Acknowledgements.

I acknowledge the financial support of the Simons Foundation (Grant No.
454949, Giorgio Parisi). I thank E. Zaccarelli for substantial help with
computing resources.

##  Appendix A Path Integral Expression of the Rate

###  A.1 Replicas

The logarithm in the quenched average can be eliminated by the Replica method

$\sum_{\sigma,\tau}P_{eq}(\sigma)P_{eq}(\tau)\,\ln\hat{T}(\sigma,\tau)\ ,$
(101)

furthermore we consider systems with quenched disorder whose equilibrium
properties can also be studied by the replica method, thus we introduce the
following object:

$Z_{\hat{T}}(m,m^{\prime},n)\equiv\sum_{\sigma_{1}\dots\sigma_{m}}\sum_{\tau_{1}\dots\tau_{m^{\prime}}}\exp\left[-\beta\sum_{i=1}^{m}H_{J}(\sigma_{i})-\beta\sum_{i=1}^{m^{\prime}}H_{J}(\tau_{i})\right]\hat{T}(\sigma_{1},\tau_{1})^{n}$
(102)

and we have:

$\sum_{\sigma,\tau}P_{eq}(\sigma)P_{eq}(\tau)\,\ln\hat{T}(\sigma,\tau)=\lim_{m\rightarrow
0}\lim_{m^{\prime}\rightarrow 0}\lim_{n\rightarrow 0}{d\over dn}\ln
Z_{\hat{T}}(m,m^{\prime},n)\ .$ (103)

As usual in the context of mean-field models the thermodynamic limit is taken
before the above limits. Note that, the quantity $\lim_{m\rightarrow
0,m^{\prime}\rightarrow 0}\,Z_{\hat{T}}(m,m^{\prime},n)$ at finite $n$ allows
to study the large deviations of $\ln\hat{T}(\sigma,\tau)$ and determine if it
is self-averaging with respect to the equilibrium configurations $\sigma$ and
$\tau$. As we will see in the next subsection it is possible to obtain an
integral representation of the dynamics (see Zinn-Justin (2002), chapter 17):

$\hat{T}(\sigma_{1},\tau_{1})=\int_{s(-\mathcal{T})=\sigma_{1},s(\mathcal{T})=\tau_{1}}ds\exp\left[{1\over
2}\int d1d2s(1)\Gamma(1,2)s(2)-\beta\int d1H_{J}(s(1))\right]\ .$ (104)

Where the $1$ and $2$ are coordinates that collect a time variable and
Fermionic variables introduced to obtain the integral representation. The
expression is made of two parts: a universal dynamic one encoded by the matrix
$\Gamma(1,2)$ and an interaction part that depends on the model encoded in the
Hamiltonian $H_{J}$. The above expressions leads to:

$\displaystyle Z_{\hat{T}}(m,m^{\prime},n)$ $\displaystyle=$
$\displaystyle\sum_{\sigma_{1}\dots\sigma_{m}}\sum_{\tau_{1}\dots\tau_{m^{\prime}}}\prod_{a=1}^{n}\left[\int_{s_{a}(-\mathcal{T})=\sigma_{1},s_{a}(\mathcal{T})=\tau_{1}}[ds_{a}]\right]\exp\left[\sum_{a=1}^{n}{1\over
2}\int d1d2s_{a}(1)\Gamma(1,2)s_{a}(2)+\right.$ (105)
$\displaystyle\left.-\beta\sum_{i=1}^{m}H_{J}(\sigma_{i})-\beta\sum_{i=1}^{m^{\prime}}H_{J}(\tau_{i})-\beta\sum_{a=1}^{n}\int
d1H_{J}(s_{a}(1))\right]\ ,$

this in turn can be written in a compact form as:

$Z_{\hat{T}}(m,m^{\prime},n)=\int[d{\bf s}]\exp\left[{1\over 2}\int{\bf
d1d2s(1)\Gamma(1,2)s(2)}-\beta\int d{\bf 1}H_{J}({\bf s(1)})\right]$ (106)

where the bold index ${\bf 1}$ runs over the replicas and the dynamical
indexes

${\bf
s(1)}=\\{\sigma_{1},\dots,\sigma_{m},s_{1}(1),\dots,s_{n}(1),\tau_{1},\dots,\tau_{m^{\prime}}\\}$
(107)

and we have:

$\int f({\bf s(1)})d{\bf
1}=\sum_{i=1}^{m}f(\sigma_{i})+\sum_{i=1}^{m^{\prime}}f(\tau_{i})+\sum_{a=1}^{n}\int
d1f(s_{a}(1))\ .$ (108)

The dynamical operator ${\bf\Gamma(1,2)}$ is diagonal with respect to the
replica indexes and associates each dynamical replica with the boundary
conditions at $\sigma_{1}$ and $\tau_{1}$. Its definition in the specific
representation we will use is given in eq. 161. As usual the great advantage
of having the above compact representation is that one can perform the average
of the disorder and then perform standard manipulations yielding an expression
formally identical to the one obtained in the case of a static replica
computation.

###  A.2 Path Integral Representation of Langevin Dynamics

This section discusses the path integral representation of Langevin dynamics,
given in terms of a single component $q$. The one-dimensional case is
discussed for simplicity, the generalisation to $N$-dimensional vectors used
in the paper is straightforward. The Langevin equation reads

${1\over\Gamma_{0}}\dot{q}=-\beta{dH\over dq}+\xi\,,\
\langle\xi(t)\xi(t^{\prime})\rangle={2\over\Gamma_{0}}\delta(t-t^{\prime})$
(109)

and we discretize it as:

${1\over\Gamma_{0}}{q_{i+i}-q_{i}\over\Delta t}=-\beta\left(c\,{dH\over
dq_{i}}+(1-c){dH\over dq_{i+1}}\right)+\xi_{i}$ (110)

where $0\leq c\leq 1$ is an arbitrary constant: in Itô discretization with
have $c=1$, in Stratonovich we have $c=1/2$. Enforcing the equations through
an integral representation we can write the average over trajectories at fixed
initial and final conditions as an integral

$T(\sigma|\tau)=\int\left(\prod_{i=1}^{N-1}dq_{i}\right)\left(\prod_{i=0}^{N-1}{d\hat{q}_{i}\over
2\pi}\right)\times$
$\times\exp\left[-\sum_{i=0}^{N-1}\left(\hat{q}_{i}E_{i}\Delta
t-\ln\left[{1\over\Gamma_{0}}-(1-c)\beta{d^{2}H\over dq_{i+1}^{2}}\,\Delta
t\right]\right)\right]$

where the interval between the initial and final time is divided in $N$ sub-
intervals of size $\Delta t$ and:

$E_{i}\equiv{1\over\Gamma_{0}}{q_{i+i}-q_{i}\over\Delta
t}+\beta\left(c\,{dH\over dq_{i}}+(1-c){dH\over dq_{i+1}}\right)-\xi_{i}$
(111)

The logarithm comes from the determinant of the Jacobian that gets
contribution only from the diagonal since the Jacobian is a triangular matrix.
Expanding the Jacobian at first order in $\Delta t$ we obtain:

$T(\sigma|\tau)=\int\left(\prod_{i=1}^{N-1}dq_{i}\right)\left(\prod_{i=0}^{N-1}{d\hat{q}_{i}\over
2\pi\Gamma_{0}}\right)\exp[-\mathcal{L}]$ (112)

The Lagrangian reads:

$\mathcal{L}=\sum_{i=0}^{N-1}\left(\hat{q}_{i}E_{i}\Delta
t-(1-c)\beta{d^{2}H\over dq_{i+1}^{2}}\,\Gamma_{0}\,\Delta t\right)$ (113)

and in the continuum limit we have

${\mathcal{L}}=\int
dt\left[{1\over\Gamma_{0}}\left(\hat{q}\dot{q}-\hat{q}^{2}\right)+\hat{q}\beta{dH\over
dq}-(1-c)\Gamma_{0}\beta{d^{2}H\over dq^{2}}\right]\ .$ (114)

Note that unexpectedly the continuum limit expression depends on the
microscopic parameter $c$ of the discretization while one would expect it to
be irrelevant. This is a well-known ambiguity of path integral representation
of stochastic equations. One can choose to use the Itô discretization
corresponding to $c=1$ and neglect it but it will resurface later in the
computation. In the following we prefer to keep it also to remind us that the
continuum limit of stochastic equations must be taken with care as the
ordinary rules of calculus (integration by parts, differentiation, chain
rules) are modified. Beside we will use Hamiltonians where the interaction
part is just linear (the $p$-spin interactions) and thus in the end we will go
back to special Langevin equation for a single variable. Let us consider the
symmetric rate defined as

$\hat{T}(\sigma,\tau)\equiv T(\sigma|\tau)e^{{\beta\over
2}\left(H(\sigma)-H(\tau)\right)}$ (115)

In the continuum limit we would expect the following to be an equality

${\beta\over 2}\left(H(\sigma)-H(\tau)\right)\neq{\beta\over
2}\int_{-\mathcal{T}}^{+\mathcal{T}}dt\,\dot{q}{\partial H\over dq}\ .$ (116)

Instead in order to get the correct expression we should go back to the
discretized expression. We have:

$H(q_{i+1})-H(q_{i})={dH\over dq_{i}}\Delta q_{i}+{1\over 2}{d^{2}H\over
dq_{i}^{2}}\Delta q_{i}^{2}$ (117)

then we have to use the fact that in the Lagrangian we use the following
discretized definition of $dH/dq$

$c\,{dH\over dq_{i}}+(1-c){dH\over dq_{i+1}}\ .$ (118)

By rewriting the differential as

${dH\over dq_{i}}=\left(c\,{dH\over dq_{i}}+(1-c){dH\over
dq_{i+1}}\right)-(1-c){d^{2}H\over dq_{i}^{2}}\Delta q_{i}$ (119)

we obtain

$dH={dH\over dq}dq+\left(c-{1\over 2}\right){d^{2}H\over dq^{2}}dq^{2}\ .$
(120)

We can see that for $c=1$ we recover Itô’s lemma while for the Stratonovich
prescription $c=1/2$ we find that the ordinary chain rule applies. The second
term cannot be neglected because it gives an $O(dt)$ contribution but we can
make the replacement

$dq^{2}=2\,\Gamma_{0}\,dt$ (121)

and obtain

${\beta\over 2}\left(H(\sigma)-H(\tau)\right)={\beta\over
2}\int_{-\mathcal{T}}^{+\mathcal{T}}dt\left({dH\over
dq}\dot{q}+\left(c-{1\over 2}\right){d^{2}H\over dq^{2}}2\,\Gamma_{0}\right)\
.$

The same result can also be obtained reabsorbing the term $\Delta q^{2}$ in
the term $\dot{q}^{2}$ below (see e.g. Zinn-Justin (2002), section 4.6).
Making the following change of variable

$\hat{q}=\hat{x}+{\dot{q}\over 2}$ (122)

we finally obtain

$\hat{\mathcal{L}}(t)={1\over\Gamma_{0}}\left({\dot{q}^{2}\over
4}-\hat{x}^{2}\right)+\hat{x}\beta{dH\over dq}-{1\over
2}\Gamma_{0}\beta{d^{2}H\over dq^{2}}\ .$ (123)

We can further integrate out the $\hat{x}$:

$\hat{\mathcal{L}}^{\prime}(t)={1\over\Gamma_{0}}{\dot{q}^{2}\over
4}+{\Gamma_{0}\over 4}\left(\beta{dH\over dq}\right)^{2}-{1\over
2}\Gamma_{0}\beta{d^{2}H\over dq^{2}}\ ,$ (124)

see also Zinn-Justin (2002), pag. 70. Note that the integration over $\hat{x}$
leads to a divergent prefactor to the above path integral:

${1\over 2\pi\Gamma_{0}}\int d\hat{x}e^{\hat{x}^{2}\over\Gamma_{0}}dt={1\over
2\pi\Gamma_{0}}\left(2\pi{\Gamma_{0}\over 2\,dt}\right)^{1/2}\ .$ (125)

which is usually buried into the expression $[dq]$ defined as:

$[dq]\equiv dq\,\left({1/(2\Gamma_{0})\over 2\pi dt}\right)^{N/2}$ (126)

see eq. 2.20 in Zinn-Justin (2002).

###  A.3 Path Integral Representation for Models with Multi-Linear
Interactions

Typical mean-field SG models have interactions that are multi-linear, while
the non-linear part of the Hamiltonian is local and does not need to be
decoupled in order to obtain a saddle point expression. This allows to use
simplified integral representations of the dynamics in which fewer variables
are introduced with respect to the general case discussed in Zinn-Justin
(2002), chapter 17. In the following we introduce a bosonic variable $\eta$
that behaves as the product of two Grassmann variables, that is we have:

$\eta^{2}=0\,,\ \int d\eta=0\,,\ \int\eta\,d\eta=1$ (127)

with it we define a new coordinate $1\equiv(t,\eta)$ and field $s(1)$:

$s_{i}(1)\equiv s_{i}(t)+\hat{x}_{i}(t)\eta\ .$ (128)

With the above definitions the interacting part of the multi-linear
Hamiltonian can be written as:

$\beta\int dt\sum_{i}\hat{x}_{i}(t){dH\over ds_{i}}=\beta\int d1H(s(1))\ .$
(129)

A similar formulation is also useful in the spherical model. In this case
however the non-linear part of the Hamiltonian is due to the spherical
constraint which is not local and must be treated appropriately. Let us
consider the problem of the integral representation of Langevin dynamics of
$N$ real spins $s_{i}$ constrained on a $(N-1)$-dimensional surface specified
by some condition $G(s)=0$. The statics of the problem can be written as

$\int d^{N}s\,\delta(G)|\nabla G|e^{-\beta H(s)}$ (130)

A convenient way to define Langevin dynamics on the surface is to relax the
delta function replacing it with a Gaussian of infinitesimal variance
$\epsilon$. Then one have to compute the standard dynamical integral in
presence of a Hamiltonian $H(x)=G^{2}(x)/(2N\epsilon)$. For the spherical
constraint on $N$ continuous spins $s_{i}$ we have

$G(s)\equiv\sum_{i=1}^{N}s_{i}^{2}-N$ (131)

the term $|\nabla G|$ is exactly equal to $N$ and can be ignored, For
$H(x)=G^{2}(x)/(2N\epsilon)$ we have:

$\sum_{i}\hat{x}_{i}\beta{dH\over dq_{i}}-{1\over
2}\Gamma_{0}\beta\sum_{i}{d^{2}H\over dq_{i}^{2}}={\beta\over
N\epsilon}G(s)\left(2\sum_{i}\hat{x}_{i}s_{i}-\Gamma_{0}N\right)$ (132)

plus $o(N)$ terms. The expression can be decoupled through a Hubbard-
Stratonovich transformation in terms of two additional fields $\mu(t)$ and
$\tilde{\mu}(t)$ and taking the limit $\epsilon\rightarrow 0$ the quadratic
part disappears leading to the following contribution to the action:

$-{1\over 2}\int dt\hat{\mu}(t)\left(\sum_{i}s_{i}^{2}-N\right)-{1\over 2}\int
dt{\mu}(t)\left(2\sum_{i}\hat{x}_{i}s_{i}-\Gamma_{0}N\right)\ .$

Introducing the variable $\mu(1)=\mu(t)+\hat{\mu}(t)\eta$ we can write the
above term as

$-{1\over 2}\int d1\mu(1)\left(\sum_{i}s_{i}^{2}(1)-N\right)+{1\over
2}\Gamma_{0}\int dt\mu(t)$

Note that presence on the second term that is essential to get the correct
saddle-point equations and the correct value of the rate.

##  Appendix B The $p$-spin Spherical Model: The Action and Saddle-Point
Equations

In the standard treatment Mézard et al. (1987) the expression of the free
energy of the fully-connected spin-glass models is obtained through a number
of steps. First $n$ replicas of the model are introduced, then the disorder
average over the partition function of the replicated system is performed and
this leads to an expression in which different replicas are coupled. Then a
Hubbard-Stratonovich transformation is performed leading to an action
depending on a single local replicated spin variable and on two $n\times n$
matrices $Q_{ab}$ and $\Lambda_{ab}$. Finally the action has to be integrated
over $Q_{ab}$ and $\Lambda_{ab}$ but the mean-field nature of the model allows
to use the saddle-point method. The very same manipulations can be applied to
the interaction part of expression (106), the only difference is that instead
of having spins $s_{a}$ labeled by indexes $a=1,\dots,n$ the spins depend on a
more complex index (or coordinate) ${\bf 1}$ specified in eq. (107). As a
result the objects $Q$ and $\Lambda$ are (formally) matrices in this more
complex coordinate. In spherical models the spin-variables can be also
integrated out and we obtain the following expression for $Z_{\hat{T}}$ of the
$p$-spin spherical model:

$\exp\,N\left[{\beta^{2}\over 4}\int d{\bf a}d{\bf b}f({\bf Q(a,b)})+{1\over
2}{\bf\Lambda(a,b)Q(a,b)}-{1\over 2}{\rm
Tr}\,\ln({\bf\Gamma}+{\bf\mu}+{\bf\Lambda})+const.+{1\over 2}\int d{\bf
a}\,\mathbf{\mu(a)}+\sum_{a=1}^{n}{\Gamma_{0}\over 2}\int\mu_{a}(t)dt\right]$
(133)

The expression $const.$ above collects a number of terms coming from the
explicit Gaussian integration, it is divergent and cancels the divergences
associated to the expression ${\rm
Tr}\,\ln({\bf\Gamma}+{\bf\mu}+{\bf\Lambda})$. These are pathologies of the
path integral representation that are fixed going back to the discrete times
and will be further discussed in appendix G. Note that the application of the
standard manipulations to the form (106) would lead to the above expression
without the last term that appears instead if we want to use the simplified
formulation which is suitable for multi-linear interactions. The above
expression has to be extremized with respect to ${\bf Q(ab)}$,
${\bf\Lambda(ab)}$ and with respect to $\bm{\mu(a)}$. Extremization with
respect to ${\bf Q(ab)}$, $\bm{\Lambda(ab)}$ leads to the saddle point
equations:

$\displaystyle{\bf\Lambda(ab)}$ $\displaystyle=$
$\displaystyle-{\beta^{2}\over 2}f^{\prime}({\bf Q(ab)})$ (134)
$\displaystyle{\bf Q}$ $\displaystyle=$
$\displaystyle{1\over\bm{\Gamma+\Lambda+\mu}}$ (135)

The last equation can be rewritten as:

$\bm{\Gamma(ac)Q(cb)+\bm{\mu(a)\,\delta(ac)Q(cb)}+\Lambda(ac)Q(cb)}=\bm{\delta(ab)}$
(136)

where there integration over the variable $\bm{c}$ is implicit. The above
expression is extremely compact, in the following we wil see that it encodes
eight integro-differential equations.

From now on we specialize to the case of a 1RSB transition. We start noticing
that the initial and final configuration are weighted with the equilibrium
Gibbs measure and their properties must not depend on the dynamics. This is
granted by the fact that the terms depending on the dynamics in the equations
for the replicas of the initial and final configurations are $O(n)$ and
disappear in the $n\rightarrow 0$ limit (that must be taken first). This means
that $Q(ab)$ with $a$ and $b$ corresponding to the $a,b=1,\dots,m+m^{\prime}$
replicas associated to the equilibrium boundary conditions is the ordinary
equilibrium replica matrix. For simplicity we will work in zero field and zero
random field meaning that:

$f^{\prime}(0)=0\,,$ (137)

in this way the overlap between different equilibrium configurations is zero.
Therefore above $T_{d}$ the solution is $Q(ab)=\delta(ab)$ that plugged into
the above equations leads to:

$\mu=1+{\beta^{2}\over 2}f^{\prime}(1)\ .$ (138)

Below $T_{d}$ the solution is actually 1RSB. The replicas determining the
boundary conditions should naturally belong to different RSB blocks ensuring
that we are studying the transition between different states. On the other
hand the equations for the dynamical part will have a non-vanishing
correlation with the remaining $x-1$ replicas in the block of the initial
configuration and with the $x-1$ replicas in the block of the final
configuration. This will lead to a correction of order $x-1$ to the dynamical
equations valid at high temperature. However for $T_{s}<T<T_{d}$ we have
exactly $x=1$ and thus this contribution vanishes. Therefore we will safely
use the high temperature Replica-Symmetric (RS) solution $Q(ab)=\delta(ab)$
also in the region $T_{s}<T<T_{d}$. This is consistent with the known result
that annealed and quenched averages are equivalent above $T_{s}$.

The global order parameter $\bm{Q(ab)}$ can be divided into a static part
corresponding to the $m+m^{\prime}$ replicas, a dynamic part describing the
$n$ replicas of the dynamics and a mixed part. We have seen before that, as it
should, the static part does not depend on the dynamics part due to the
$n\rightarrow 0$ limit. We now focus on the dynamic part. We will make a RS
ansatz on the $n$ dynamical replicas, therefore the dynamical component of
$\bm{Q(ab)}$ will be characterized by two matrices

$\bm{Q(ab)}=\delta_{\alpha\beta}Q(ab)+(1-\delta_{\alpha\beta})Q^{dt}(ab)$
(139)

where we have moved from the full coordinates $\bm{(ab)}$ to purely dynamical
coordinates $(ab)$ and replica coordinates $\alpha,\beta=1,\dots,n$. The RS
ansatz implies also

$\mu_{\alpha}(t)=\mu(t)\,,\ \hat{\mu}_{\alpha}(t)=\hat{\mu}(t)\,$ (140)

and we have:

$\displaystyle\bm{\Gamma(ab)}$ $\displaystyle=$
$\displaystyle\delta_{\alpha\beta}\Gamma(ab)$ (141)
$\displaystyle\bm{\delta(ab)}$ $\displaystyle=$
$\displaystyle\delta_{\alpha\beta}\delta(ab)\ .$ (142)

The dynamical components of equations (136) can then be rewritten in a form
that can be analitically continued to real values of the replica number $n$

$\Gamma(ac)Q(cb)+\beta\mu(a)\delta(ac)Q(cb)+(\Lambda Q)^{st}=\delta(ab)$ (143)
$\Gamma(ac)Q^{dt}(cb)+\beta\mu(a)\delta(ac)Q^{dt}(cb)+(\Lambda Q)^{dt}=0$
(144)

where again the integration over the variable $c$ is implicit and

$\displaystyle(\Lambda Q)^{st}$ $\displaystyle=$
$\displaystyle\Lambda(ac)Q(cb)+(n-1)\Lambda^{dt}(ac)Q^{dt}(cb)+$ (145)
$\displaystyle+$ $\displaystyle\Lambda(a1)Q(1b)+\Lambda(a2)Q(2b)\ ^{\prime}$
$\displaystyle(\Lambda Q)^{dt}$ $\displaystyle=$
$\displaystyle\Lambda^{dt}(ac)Q(cb)+\Lambda(ac)Q^{dt}(cb)+$ (146)
$\displaystyle+$ $\displaystyle(n-2)\Lambda^{dt}(ac)Q^{dt}(cb)+$
$\displaystyle+$ $\displaystyle\Lambda(a1)Q(1b)+\Lambda(a2)Q(2b)\ ,$

where $1$ and $2$ label the two static replicas whose configurations are
chosen as initial and final condition of the dynamics at times
$\mp\mathcal{T}$. Note that the corresponding terms appear when we integrate
over the full coordinate ${\bm{c}}$ in eq. (136). In the above equations we
have naturally:

$\displaystyle\Lambda(ab)$ $\displaystyle=$ $\displaystyle-{\beta^{2}\over
2}f^{\prime}(Q(ab))$ (147) $\displaystyle\Lambda^{dt}(ab)$ $\displaystyle=$
$\displaystyle-{\beta^{2}\over 2}f^{\prime}(Q^{dt}(ab))\ .$ (148)

$Q(ab)$ and $Q^{dt}(ab)$ can be expressed in terms of four two-time functions
as:

$\displaystyle Q(ab)$ $\displaystyle\equiv$ $\displaystyle
C(t_{a},t_{b})+\hat{R}_{1}(t_{a},t_{b})\eta_{a}+\hat{R}_{2}(t_{a},t_{b})\eta_{b}+$
(149) $\displaystyle+$ $\displaystyle\hat{X}(t_{a},t_{b})\eta_{a}\eta_{b}$
$\displaystyle Q^{dt}(ab)$ $\displaystyle\equiv$ $\displaystyle
C^{dt}(t_{a},t_{b})+\hat{R}^{dt}_{1}(t_{a},t_{b})\eta_{a}+\hat{R}^{dt}_{2}(t_{a},t_{b})\eta_{b}+$
(150) $\displaystyle+$
$\displaystyle\hat{X}^{dt}(t_{a},t_{b})\eta_{a}\eta_{b}$

The same representation can be obtained from any function $A(ab)$ as

$\displaystyle A(ab)$ $\displaystyle\equiv$ $\displaystyle
C_{A}(t_{a},t_{b})+\hat{R}_{1,A}(t_{a},t_{b})\eta_{a}+\hat{R}_{2,A}(t_{a},t_{b})\eta_{b}+$
(151) $\displaystyle+$ $\displaystyle\hat{X}_{A}(t_{a},t_{b})\eta_{a}\eta_{b}\
.$

from which we obtain

$\displaystyle C_{\Lambda}(t_{a},t_{b})$ $\displaystyle=$
$\displaystyle-{\beta^{2}\over 2}f^{\prime}(C(t_{a},t_{b}))$ (152)
$\displaystyle\hat{R}_{1,\Lambda}(t_{a},t_{b})$ $\displaystyle=$
$\displaystyle-{\beta^{2}\over
2}f^{\prime\prime}(C(t_{a},t_{b}))\,\hat{R}_{1}(t_{a},t_{b})$ (153)
$\displaystyle\hat{R}_{2,\Lambda}(t_{a},t_{b})$ $\displaystyle=$
$\displaystyle-{\beta^{2}\over
2}f^{\prime\prime}(C(t_{a},t_{b}))\,\hat{R}_{2}(t_{a},t_{b})$ (154)
$\hat{X}_{\Lambda}(t_{a},t_{b})=-{\beta^{2}\over
2}f^{\prime\prime}(C(t_{a},t_{b}))\,\hat{X}(t_{a},t_{b})+$ $-{\beta^{2}\over
2}f^{\prime\prime\prime}(C(t_{a},t_{b}))\,\hat{R}_{1}(t_{a},t_{b})\hat{R}_{2}(t_{a},t_{b})$
(155) $C_{\Lambda^{dt}}(t_{a},t_{b})=-{\beta^{2}\over
2}f^{\prime}(C^{dt}(t_{a},t_{b}))$ (156)
$\hat{R}_{1,\Lambda^{dt}}(t_{a},t_{b})=-{\beta^{2}\over
2}f^{\prime\prime}(C^{dt}(t_{a},t_{b}))\,\hat{R}^{dt}_{1}(t_{a},t_{b})$ (157)
$\hat{R}_{2,\Lambda^{dt}}(t_{a},t_{b})=-{\beta^{2}\over
2}f^{\prime\prime}(C^{dt}(t_{a},t_{b}))\,\hat{R}^{dt}_{2}(t_{a},t_{b})$ (158)
$\hat{X}_{\Lambda^{dt}}(t_{a},t_{b})=-{\beta^{2}\over
2}f^{\prime\prime}(C^{dt}(t_{a},t_{b}))\,\hat{X}^{dt}(t_{a},t_{b})+$
$-{\beta^{2}\over
2}f^{\prime\prime\prime}(C^{dt}(t_{a},t_{b}))\,\hat{R}^{dt}_{1}(t_{a},t_{b})\hat{R}^{dt}_{2}(t_{a},t_{b})\
.$ (159)

The operator $\Gamma(ab)$ is defined through

${1\over 2}\int
dadb\,\Gamma(ab)q(a)q(b)\equiv\int{1\over\Gamma_{0}}\left({\dot{q}^{2}\over
4}-\hat{x}^{2}\right)dt\ ,$ (160)

that leads to

$\Gamma(ab)\equiv{1\over\Gamma_{0}}\left[-{1\over
2}\delta^{\prime\prime}(t_{a}-t_{b})\eta_{a}\,\eta_{b}-2\,\delta(t_{a}-t_{b})\right]\
.$ (161)

For $\Gamma_{0}=1$ we have:

$\displaystyle\Gamma(ac)Q(cb)$ $\displaystyle=$
$\displaystyle-2\hat{R}_{1}(t_{a},t_{b})-{1\over 2}{d^{2}\over
dt_{a}^{2}}C(t_{a},t_{b})\eta_{a}+$ $\displaystyle-$ $\displaystyle
2\hat{X}(t_{a},t_{b})\eta_{b}-{1\over 2}{d^{2}\over
dt_{a}^{2}}\hat{R}_{2}(t_{a},t_{b})\eta_{a}\eta_{b}\ \ .$

The corresponding expression in eq. (144) is obtained replacing $Q(ab)$ with
$Q^{dt}(ab)$. The term depending on $\mu$ can be written as:

$\displaystyle\mu(a)Q(ab)$ $\displaystyle=$
$\displaystyle\mu(t_{a})C(t_{a},t_{b})+$ $\displaystyle+$
$\displaystyle\eta_{a}\left(\mu(t_{a})\hat{R}_{1}(t_{a},t_{b})+\hat{\mu}(t_{a})C(t_{a},t_{b})\right)+$
$\displaystyle+$ $\displaystyle\eta_{b}\,\mu(t_{a})\hat{R}_{2}(t_{a},t_{b})+$
$\displaystyle+$
$\displaystyle\eta_{a}\eta_{b}\left(\hat{\mu}(t_{a})\hat{R}_{2}(t_{a},t_{b})+\mu(t_{a})\hat{X}(t_{a},t_{b})\right).$

As above, the corresponding expression in eq. (144) is obtained replacing
$Q(ab)$ with $Q^{dt}(ab)$. In order to complete the derivation of the saddle-
point equations we need the expression for a product of the form $A(ac)B(cb)$:

$\displaystyle A(ac)B(cb)$ $\displaystyle=$ $\displaystyle
C_{A}(t_{a},t_{c})\hat{R}_{1,B}(t_{c},t_{b})+\hat{R}_{2,A}(t_{a},t_{c})C_{B}(t_{c},t_{b})+\eta_{a}\left(\hat{R}_{1,A}(t_{a},t_{c})\hat{R}_{1,B}(t_{c},t_{b})+\hat{X}_{A}(t_{a},t_{c})C_{B}(t_{c},t_{b})\right)+$
(162) $\displaystyle+$
$\displaystyle\eta_{b}\left(\hat{R}_{2,A}(t_{a},t_{c})\hat{R}_{2,B}(t_{c},t_{b})+C_{A}(t_{a},t_{c})\hat{X}_{B}(t_{c},t_{b})\right)+$
$\displaystyle+$
$\displaystyle\eta_{a}\eta_{b}\left(\hat{R}_{1,A}(t_{a},t_{c})\hat{X}_{B}(t_{c},t_{b})+\hat{X}_{A}(t_{a},t_{c})\hat{R}_{2,B}(t_{c},t_{b})\right)\
.$

For the contributions of the initial and final configurations in the
interaction term we have:

$\displaystyle\Lambda(a1)Q(1b)+\Lambda(a2)Q(2b)$ $\displaystyle=$
$\displaystyle
C_{\Lambda}(t_{a},-\mathcal{T})C(-\mathcal{T},t_{b})+C_{\Lambda}(t_{a},\mathcal{T})C(\mathcal{T},t_{b})$
(163) $\displaystyle+$
$\displaystyle\eta_{a}\left(\hat{R}_{1,\Lambda}(t_{a},-\mathcal{T})C(-\mathcal{T},t_{b})+\hat{R}_{1,\Lambda}(t_{a},\mathcal{T})C(\mathcal{T},t_{b})\right)+$
$\displaystyle+$
$\displaystyle\eta_{b}\left(C_{\Lambda}(t_{a},-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t_{b})+C_{\Lambda}(t_{a},\mathcal{T})\hat{R}_{2}(\mathcal{T},t_{b})\right)+$
$\displaystyle+$
$\displaystyle\eta_{a}\eta_{b}\left(\hat{R}_{1,\Lambda}(t_{a},-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t_{b})+\hat{R}_{1,\Lambda}(t_{a},\mathcal{T})\hat{R}_{2}(\mathcal{T},t_{b})\right)\
.$

We also have:

$\delta(ab)=(\eta_{a}+\eta_{b})\delta(t_{a}-t_{b})\ .$ (164)

Collecting the various components in eq. (143) and (144) we obtain eight
integro-differential equations that will be written in expanded form in
appendix C and that were written in compact form in section II.2. The
extremization of expression (133) with respect to $\mu(t)$ and $\hat{\mu}(t)$
gives the following conditions:

$C(t,t,)=1\,,\ \ \hat{R}_{1}(t,t)=\hat{R}_{2}(t,t^{\prime})={1\over 2}\ .$
(165)

note that the last term in (133) is essential to obtain the correct expression
for $\hat{R}_{1}(t,t)$ and $\hat{R}_{2}(t,t)$

##  Appendix C The equations in expanded form

Expressions (45) and (46) are compact and useful for a numerical treatment.
They correspond to eight integro-differential equations that we write in the
following in explicit form for completeness. They have to be supplemented with
the definitions (152-159) and the boundary conditions discussed in section
II.2.

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\hat{R}_{1}(t,t^{\prime})+\mu(t)C(t,t^{\prime})+$ (166)
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(C_{\Lambda}(t,t^{\prime\prime})\hat{R}_{1}(t^{\prime\prime},t^{\prime})+\hat{R}_{2,\Lambda}(t,t^{\prime\prime})C(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{R}_{2,{\Lambda^{dt}}}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$ $\displaystyle
C_{\Lambda}(t,-\mathcal{T})C(-\mathcal{T},t^{\prime})+C_{\Lambda}(t,\mathcal{T})C(\mathcal{T},t^{\prime})\
.$

$\displaystyle\delta(t-t^{\prime})$ $\displaystyle=$ $\displaystyle-{1\over
2}{d^{2}\over
dt^{2}}C(t,t^{\prime})+\mu(t)\hat{R}_{1}(t,t^{\prime})+\hat{\mu}(t)C(t,t^{\prime})+$
(167) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda}(t,t^{\prime\prime})\hat{R}_{1}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda}(t,t^{\prime\prime})C(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\hat{R}_{1,\Lambda}(t,-\mathcal{T})C(-\mathcal{T},t^{\prime})+\hat{R}_{1,\Lambda}(t,\mathcal{T})C(\mathcal{T},t^{\prime})\
.$

$\displaystyle\delta(t-t^{\prime})$ $\displaystyle=$
$\displaystyle-2\hat{X}(t,t^{\prime})+\mu(t)\hat{R}_{2}(t,t^{\prime})+$ (168)
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{2,\Lambda}(t,t^{\prime\prime})\hat{R}_{2}(t^{\prime\prime},t^{\prime})+C_{\Lambda}(t,t^{\prime\prime})\hat{X}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{2,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})+C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$ $\displaystyle
C_{\Lambda}(t,-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t^{\prime})+C_{\Lambda}(t,\mathcal{T})\hat{R}_{2}(\mathcal{T},t^{\prime})\
.$

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}\hat{R}_{2}(t,t^{\prime})+\mu(t)\hat{X}(t,t^{\prime})+\hat{\mu}(t)\hat{R}_{2}(t,t^{\prime})+$
(169) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda}(t,t^{\prime\prime})\hat{X}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda}(t,t^{\prime\prime})\hat{R}_{2}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\hat{R}_{1,\Lambda}(t,-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t^{\prime})+\hat{R}_{1,\Lambda}(t,\mathcal{T})\hat{R}_{2}(\mathcal{T},t^{\prime})\
.$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\hat{R}_{1}^{dt}(t,t^{\prime})+\mu(t)C^{dt}(t,t^{\prime})+$
(170) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}(t^{\prime\prime},t^{\prime})+\hat{R}_{2,{\Lambda^{dt}}}(t,t^{\prime\prime})C(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(C_{\Lambda}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{R}_{2,{\Lambda}}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-2)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{R}_{2,{\Lambda^{dt}}}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$ $\displaystyle
C_{\Lambda}(t,-\mathcal{T})C(-\mathcal{T},t^{\prime})+C_{\Lambda}(t,\mathcal{T})C(\mathcal{T},t^{\prime})\
.$

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}C^{dt}(t,t^{\prime})+\mu(t)\hat{R}_{1}^{dt}(t,t^{\prime})+\hat{\mu}(t)C^{dt}(t,t^{\prime})+$
(171) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})C(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-2)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{1}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})C^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\hat{R}_{1,\Lambda}(t,-\mathcal{T})C(-\mathcal{T},t^{\prime})+\hat{R}_{1,\Lambda}(t,\mathcal{T})C(\mathcal{T},t^{\prime})\
.$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\hat{X}^{dt}(t,t^{\prime})+\mu(t)\hat{R}_{2}^{dt}(t,t^{\prime})+$
(172) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{2,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}(t^{\prime\prime},t^{\prime})+C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{2,\Lambda}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})+C_{\Lambda}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-2)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{2,\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})+C_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$ $\displaystyle
C_{\Lambda}(t,-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t^{\prime})+C_{\Lambda}(t,\mathcal{T})\hat{R}_{2}(\mathcal{T},t^{\prime})\
.$

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-{1\over 2}{d^{2}\over
dt^{2}}\hat{R}_{2}^{dt}(t,t^{\prime})+\mu(t)\hat{X}^{dt}(t,t^{\prime})+\hat{\mu}(t)\hat{R}_{2}^{dt}(t,t^{\prime})+$
(173) $\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle(n-2)\int_{-\mathcal{T}}^{+\mathcal{T}}\left(\hat{R}_{1,\Lambda^{dt}}(t,t^{\prime\prime})\hat{X}^{dt}(t^{\prime\prime},t^{\prime})+\hat{X}_{\Lambda^{dt}}(t,t^{\prime\prime})\hat{R}_{2}^{dt}(t^{\prime\prime},t^{\prime})\right)dt^{\prime\prime}+$
$\displaystyle+$
$\displaystyle\hat{R}_{1,\Lambda}(t,-\mathcal{T})\hat{R}_{2}(-\mathcal{T},t^{\prime})+\hat{R}_{1,\Lambda}(t,\mathcal{T})\hat{R}_{2}(\mathcal{T},t^{\prime})\
.$

##  Appendix D The solutions on the corners and on the diagonal

In this appendix we discuss a number of useful properties that follow from the
equations. Equation 168 implies that

$\hat{X}(t,t^{\prime})=-{1\over
2}\delta(t-t^{\prime})+\delta\hat{X}(t,t^{\prime})$ (174)

where $\delta\hat{X}(t,t^{\prime})$ is a bounded function. In the limit
$t,t^{\prime}\rightarrow\pm\mathcal{T}$ the interaction part in the equations
greatly simplifies because we have

$C(-\mathcal{T},t)=C^{dt}(-\mathcal{T},t)\,,\
\hat{R}_{1}(t,-\mathcal{T})=\hat{R}_{1}^{dt}(t,-\mathcal{T})\,,$ $\
\delta\hat{X}(-\mathcal{T},t)=\hat{X}^{dt}(-\mathcal{T},t)\ .$

This allows to characterize the order parameters on the corners of the
$t,t^{\prime}$ domain. In particular one easily sees that for $n=0$ the
contributions of the integrals between $-\mathcal{T}$ and $\mathcal{T}$ cancel
in the limit $t,t^{\prime}\rightarrow-\mathcal{T}$ in all equations. In
particular eq. 166 leads to:

$-1+\mu(-\mathcal{T})-{\beta^{2}\over
2}f^{\prime}(1)=0\rightarrow\lim_{t\rightarrow\pm\mathcal{T}}\mu(t)=\mu_{eq}\
.$ (175)

Thus $\mu(t)$ goes continuously to the equilibrium value as
$t\rightarrow\pm\mathcal{T}$. Equations 168 and 172 lead to:

$\delta\hat{X}(\pm\mathcal{T},\pm\mathcal{T})=\hat{X}^{dt}(\pm\mathcal{T},\pm\mathcal{T})={\mu_{eq}\over
4}={1\over 4}+{\beta^{2}\over 8}f^{\prime}(1)$ (176)

Similarly for $n=0$ all terms depending on $\Lambda$ cancels in the limit
$t\rightarrow\mp\mathcal{T},\,\ t^{\prime}\rightarrow\pm\mathcal{T}$ due to
the boundary conditions

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
C(\pm\mathcal{T},\mp\mathcal{T})=C^{dt}(\pm\mathcal{T},\mp\mathcal{T})$ (177)

and one obtains from eqs. 166 and 168:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\hat{R}_{1}(\pm\mathcal{T},\mp\mathcal{T})=\hat{R}_{1}^{dt}(\pm\mathcal{T},\mp\mathcal{T})$
(178) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\hat{R}_{2}(\pm\mathcal{T},\mp\mathcal{T})=\hat{R}_{2}^{dt}(\pm\mathcal{T},\mp\mathcal{T})$
(179) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\delta\hat{X}(\pm\mathcal{T},\mp\mathcal{T})=\hat{X}^{dt}(\pm\mathcal{T},\mp\mathcal{T})$
(180)

##  Appendix E Derivatives of the logarithm of the rate

The expression (133) of the logarithm of the rate in terms of the order
parameter obtained in the path integral formulation is divergent as is shown
explicitly for the free case in appendix (G) and its computation requires to
go back to the discretized case. A convenient alternative is to compute the
derivative of the logarithm of the rate with respect to the temperature and
integrate the result using the knowledge of the infinite temperature limit
given in section II.3. The partial derivative with respect to the temperature
of expression (133) is simply given by ${\beta\over 2}{d\over dn}\int d{\bf
a}d{\bf b}f({\bf Q(a,b)})$ and it coincides with the total derivative when
computed on the solution of the saddle-point equations. Using the formulas for
the integrals computed below (appendix E.1) we obtain:

${1\over N}\,{\partial\overline{[\ln\hat{T}]}\over\partial\beta}={\beta\over
2}\left(4\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{R}_{1,f[C]}(t,-\mathcal{T})dt+\int_{-\mathcal{T}}^{+\mathcal{T}}\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{X}_{f[C]}(t,t^{\prime})dtdt^{\prime}+(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{X}_{f[C^{dt}]}(t,t^{\prime})dtdt^{\prime}\right)\
.$ (181)

As discussed in sec. I.1.3 it is interesting to consider a generalized
$[\ln\hat{T}]$ in which the final configuration at time $+\mathcal{T}$ is
selected with the Gibbs weight corresponding to a different temperature
$\beta_{2}$. This allows to compute the total probability to jump to states
with energy higher than the equilibrium one. Again the actual computation of
the divergent expression can be avoided studying the derivative of
$[\ln\hat{T}]$ with respect to $\beta_{2}$. The same argument above implies
that the total derivative coincides with the partial derivative. With a
computation similar to that leading to eq. (191) one finds:

${1\over
N}\,{\partial\overline{[\ln\hat{T}]}\over\partial\beta_{2}}=\frac{\beta}{2}\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{R}_{1,f[C]}(t,\mathcal{T})dt$
(182)

Note that when $\beta_{2}\neq\beta$ the order-parameter functions are no
longer symmetric with respect to the exchange $t\rightarrow-t$. The
probability of jumping to the configurations with energy $E_{2}$ (the
equilibrium energy at inverse temperature $\beta_{2}$) with typical rate is
then:

$\exp\left[\beta E(\beta)/2+\overline{[\ln\hat{T}]}-\beta
E(\beta_{2})/2+S(\beta_{2})\right]$ (183)

and the derivative of the logarithm of the above expression with respect to
$\beta_{2}$ is:

${\beta\over
2}\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{R}_{1,f[C]}(t,\mathcal{T})dt-{\beta\over
4}f(1)\ .$ (184)

I have not solved numerically the case $\beta_{2}\neq\beta$, but the
evaluation of the the derivative at $\beta_{2}=\beta$ allows to characterize
qualitatively the total logarithm of the rate for $\beta_{2}\neq\beta$ in the
neighborhood of $\beta$. This quantity turns out to be negative when evaluated
on the numerical solution at $\beta=1.695$ with $\beta_{2}=\beta$ meaning that
the total rate is larger considering configurations with energies larger than
equilibrium, which is the key result at bases of the discussion of section
(I.1.3).

Note that consistently in the ergodic phase expression (184) vanishes in the
$\mathcal{T}\rightarrow\infty$ limit because the solutions satisfy eqs. (77).
Instead, as we just saw, for $T<T_{d}$ the above quantity is negative for
$\beta_{2}=\beta$ also in the $\mathcal{T}\rightarrow\infty$ limit and should
vanish for some $\beta_{2}=\beta_{max}<\beta$ corresponding to a maximum at a
higher energy (and temperature) with $\beta_{max}\rightarrow\beta$ as
$T\rightarrow T_{d}^{-}$ .

###  E.1 Integrals

We consider the expression of the sum of the elements of a generic object
$\bm{B(ab)}$ in terms of its components,

$\int\bm{B(ab)dadb}\ .$ (185)

We specialize to the case in which $\bm{B(ab)}$ has the structure of the order
parameter $\bm{Q(ab)}$ as discussed in section B. In particular we have for
the $m+m^{\prime}$ static replicas:

$B(\alpha,\beta)=\delta_{\alpha\beta}C_{B}(\mathcal{T},\mathcal{T})$ (186)

similarly the $n$ dynamical replicas are correlated only with the static
replicas associated to the initial and final conditions and their correlations
have a RS form. We define

$\bm{A(a)}\equiv\int\bm{B(ba)db}\ .$ (187)

For $\bm{a}$ corresponding to any of the $m+m^{\prime}$ static replicas other
than those fixing the initial and final condition we simply have

$\bm{A(a)}=C_{B}(\mathcal{T},\mathcal{T})$ (188)

For $\bm{a}$ corresponding to either one of the two static replicas
controlling the initial and final conditions we have:

$\displaystyle\bm{A}(\mathcal{T})$ $\displaystyle=$
$\displaystyle\int\bm{B}(\mathcal{T},\bm{b})\bm{db}=C_{B}(\mathcal{T},\mathcal{T})+C_{B}(\mathcal{T},-\mathcal{T})+$
(189) $\displaystyle+$ $\displaystyle n\int\hat{R}_{2,B}(\mathcal{T},t)dt$

For $\bm{a}$ given by the $\eta$ component of one of the dynamical replicas we
have:

$\displaystyle\hat{a}(t)$ $\displaystyle=$
$\displaystyle\hat{R}_{1,B}(t,\mathcal{T})+\hat{R}_{1,B}(t,-\mathcal{T})+\int\hat{X}_{B}(t,t^{\prime})dt^{\prime}+$
(190) $\displaystyle+$
$\displaystyle(n-1)\int\hat{X}_{B}^{dt}(t,t^{\prime})dt^{\prime}\,.$

Putting everything together one finds:

$\displaystyle\int\bm{B(ab)dadb}$ $\displaystyle=$
$\displaystyle\int\bm{A(a)da}=(m+m^{\prime})\,C_{B}(\mathcal{T},\mathcal{T})+$
(191) $\displaystyle+$ $\displaystyle
n\left(\int\hat{R}_{1,B}(t,-\mathcal{T})dt+\int\hat{R}_{1,B}(t,\mathcal{T})dt+\int\hat{R}_{2,B}(\mathcal{T},t)dt+\int\hat{R}_{2,B}(-\mathcal{T},t)dt\right.+$
$\displaystyle+$
$\displaystyle\left.\int\hat{X}_{B}(t,t^{\prime})\,dt\,dt^{\prime}+(n-1)\int\hat{X}_{B^{dt}}(t,t^{\prime})\,dt\,dt^{\prime}\right)$

##  Appendix F The energy

The instantaneous energy on the trajectory reads:

$E(t)\equiv\sum_{p=1}^{\infty}\sum_{i_{1}<\dots<i_{p}}\overline{J_{i_{1}\dots
i_{p}}[\langle s_{i_{1}}(t)\dots s_{i_{p}}(t)\rangle]}\ .$ (192)

Exploiting the fact that the $J$’s are Gaussian random variables through an
integration by part one easily obtains:

$e(t)\equiv{E(t)\over N}=-{\beta\over
2}\left(C_{f[C]}(t,-\mathcal{T})+C_{f[C]}(t,\mathcal{T})+\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{R}_{2,f[C]}(t,t^{\prime})dt^{\prime}+(n-1)\int_{-\mathcal{T}}^{+\mathcal{T}}\hat{R}_{2,f[C^{dt}]}(t,t^{\prime})dt^{\prime}\right)$
(193)

In the special case of the pure $p$-spin i.e. $f(x)=x^{p}$ this leads to a
simple relationship between the energy and $\mu(t)$:

$e(t)={1\over p\,\beta}(1-\mu(t))\ .$ (194)

The above can be shown using equation (166) at equal times and using the
property that

$f^{\prime}(x)+f^{\prime\prime}(x)\,x=p\,f^{\prime}(x)\,,\ \mathrm{for}\ \
f(x)=x^{p}\ ,$ (195)

to make a connection with expression (193).

##  Appendix G The Transition Rate in The free case

In the free case the expression for the Replicated logarithm transition rate
(133) simplifies considerably due to $\beta=0$ and $\bm{\Lambda(ab)}=0$. In
the RS ansatz one obtains:

$n\left({1\over 2}\int dt\,\hat{\mu}(t)+{\Gamma_{0}\over 2}\int\mu(t)dt+\int\
{d\sigma\over\sqrt{2\pi}}e^{-{\sigma^{2}\over
2}}\int{d\tau\over\sqrt{2\pi}}e^{-{\tau^{2}\over 2}}\ln Z(\sigma,\tau)\right)\
,$
$Z(\sigma,\tau)=\int_{q(-\mathcal{T})=\sigma}^{q(\mathcal{T})=\tau}[dq]\exp\left[\int
dt\left(-{\dot{q}^{2}\over 4\,\Gamma_{0}}-{\Gamma_{0}\over
4}\mu^{2}q^{2}-{1\over 2}\hat{\mu}q^{2}\right)\right]\ .$ (196)

We recognize the path integral representation of the Harmonic oscillator that
is usually written as

$Z(\sigma,\tau)=\int_{q(-\mathcal{T})=\sigma}^{q(\mathcal{T})=\tau}[dq]\exp\left[-\int
dt\left({1\over 2}\,m\,\dot{q}^{2}+{1\over 2}m\omega^{2}q^{2}\right)\right]$

with the identification $m=1/(2\Gamma_{0})$ that leads to the same
$\dot{q}^{2}$ factor and the same $[dq]$. As discussed in classic textbooks
the above path integral is ill defined. This is easily seen switching to a
frequency representation where it is ultraviolet divergent as $\ln
Z\propto\int^{\infty}dk\,\ln(\omega^{2}+k^{2})$. The actual quantity
$Z(\sigma,\tau)$ is finite because the differential $[dq]$ includes a
prefactor diverging as $1/\Delta t$ as we have seen in section (A.2). A
careful computation leads to the following expression for $\ln Z(\sigma,\tau)$
(see eq. 2.23 in Zinn-Justin’s Zinn-Justin (2002) or eq. 13.45 in Parisi’s
Parisi (1988) ):

${1\over 2}\ln{m\omega\over 2\pi\sinh\omega\tau}-{m\omega\over
2\sinh\omega\tau}\left[\cosh\omega\tau(\sigma^{2}+\tau^{2})-2\sigma\tau\right]$
$m={1\over 2}\,,\ \omega=\sqrt{a}\,,\
\tau\equiv=t_{fin}-t_{in}=2\mathcal{T}\,,$

where we have fixed $\Gamma_{0}=1$ and used the results $\mu=1$ and $a\equiv
1+2\hat{\mu}$ obtained in section II.3. Performing the averages over $\sigma$
and $\tau$ we finally obtain:

${1\over N}\overline{[\ln\hat{T}]}=-{1\over 2}\ln 2\pi-{1\over
2}+(1+\hat{\mu})\mathcal{T}+{1\over 4}\ln{-\hat{\mu}\over 2}$ (197)

## References

  * Charbonneau et al. (2014) P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Nature communications 5, 1 (2014).
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