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Gibbs-non-Gibbs transitions, Ising models, Glauber dynamics, large deviations, trajectories
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\begin{document}
\title{
A large-deviation view on\\
dynamical Gibbs-non-Gibbs transitions}
\author{
\renewcommand{\thefootnote}{\arabic{footnote}}
A.C.D. van Enter
\footnotemark[1]
\\
\renewcommand{\thefootnote}{\arabic{footnote}}
R. Fern\'andez
\footnotemark[2]
\\
\renewcommand{\thefootnote}{\arabic{footnote}}
F. den Hollander
\footnotemark[3]
\\
\renewcommand{\thefootnote}{\arabic{footnote}}
F. Redig
\footnotemark[4]
}
\footnotetext[1]{
Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen,
P.O.\ Box 407, 9700 AK, Groningen, The Netherlands, {\sl A.C.D.van.Enter@rug.nl}
}
\footnotetext[2]{
Department of Mathematics, Utrecht University, P.O.\ Box 80010, 3508 TA Utrecht,
The Netherlands, {\sl R.Fernandez1@uu.nl}
}
\footnotetext[3]{
Mathematical Institute, Leiden University, P.O.\ Box 9512, 2300 RA, Leiden, The
Netherlands, \newline {\sl denholla@math.leidenuniv.nl}
}
\footnotetext[4]{
IMAPP, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The
Netherlands, {\sl f.redig@math.ru.nl}
}
\maketitle
\begin{abstract}
We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions
in spin systems subject to a stochastic spin-flip dynamics. Using the general
theory for large deviations of functionals of Markov processes outlined in Feng
and Kurtz~\cite{FeKu06}, we show that the trajectory under the spin-flip dynamics
of the empirical measure of the spins in a large block in $\Zd$ satisfies a large
deviation principle in the limit as the block size tends to infinity. The associated
rate function can be computed as the action functional of a Lagrangian that is the
Legendre transform of a certain non-linear generator, playing a role analogous to
the moment-generating function in the G\"artner-Ellis theorem of large deviation
theory when this is applied to finite-dimensional Markov processes. This rate function
is used to define the notion of ``bad empirical measures'', which are the discontinuity
points of the optimal trajectories (i.e., the trajectories minimizing the rate function)
given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs
transitions are linked to the occurrence of bad empirical measures: for short times
no bad empirical measures occur, while for intermediate and large times bad empirical
measures are possible. A future research program is proposed to classify the various
possible scenarios behind this crossover, which we refer to as a ``nature-versus-nurture''
transition.
\vskip 0.5truecm
\noindent
{\it MSC2010}: Primary 60F10, 60G60, 60K35; Secondary 82B26, 82C22.\\
{\it Key words and phrases}: Stochastic spin-flip dynamics, Gibbs-non-Gibbs transition,
empirical measure, non-linear generator, Nisio control semigroup, large deviation
principle, bad configurations, bad empirical measures, nature versus nurture.\\
{\it Acknowledgment:} The authors are grateful for extended discussions with
Christof K\"ulske. Part of this research was supported by the Dutch mathematics
cluster \emph{Nonlinear Dynamics of Natural Systems}. RF is grateful to NWO
(Netherlands) and CNRS (France) for financial support during his sabbatical leave
from Rouen University in the academic year 2008--2009, which took place at Groningen
University, Leiden University and EURANDOM. In the Fall of 2008 he was EURANDOM-chair.
\end{abstract}
%%%%%%%%%%%%% SECTION 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction, main results and research program}
\label{S1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamical Gibbs-non-Gibbs transitions}
\label{S1.1}
Since the discovery of the Griffiths-Pearce-Israel pathologies in renormalization-group
transformations of Gibbs measures, there has been an extensive effort towards
understanding the phenomenon that a simple transformation of a Gibbs measure
may give rise to a non-Gibbs measure, i.e., a measure for which no reasonable
Hamiltonian can be defined (see van Enter, Fern\'andez and Sokal~\cite{vEnFeSo93},
Fern\'andez~\cite{Fe06}, and the papers in the EURANDOM workshop proceedings
\cite{MPRF}). From the start, R.L. Dobrushin was interested and involved in
this development; indeed, Dobrushin and Shlosman~\cite{DoSh97}, \cite{DoSh99}
proposed a programme of \emph{Gibbsian restoration}, based on the idea that the
pathological \emph{bad configurations} of a transformed Gibbs measure
(i.e., the essential points of discontinuity of some of its finite-set,
e.g. single-site, conditional probabilities)
are exceptional in the measure-theoretic sense (i.e., they form a set of measure zero).
This has led to two extended notions of Gibbs measures: \emph{weakly Gibbsian}
measures and \emph{almost Gibbsian} measures (see Maes, Redig and Van
Moffaert~\cite{MaVaMoRe99}). Later, several refined notions were proposed, such
as \emph{intuitively weakly Gibbs} (Van Enter and Verbitskiy~\cite{vEnVe04})
and right-continuous conditional probabilities.
In Van Enter, Fern\'andez, den Hollander and Redig~\cite{vEnFedHoRe02}, the
behavior of a Gibbs measure $\mu$ subject to a high-temperature Glauber spin-flip
dynamics was considered. A guiding example is the case where we start from the
low-temperature plus-phase of the Ising model, and we run a high-temperature
dynamics, modeling the fast heating up of a cold system. The question of
Gibbsianness of the measure $\mu_t$ at
time $t>0$ can then be interpreted as the existence of a reasonable notion of an
\emph{intermediate-time-dependent temperature} (at time $t=0$ the temperature is
determined by the choice of the initial Gibbs measure, while at time $t=\infty$
the temperature is determined by the unique stationary measure of the dynamics).
For infinite-temperature dynamics, the effect of the dynamics is simply that of
a single-site Kadanoff transformation, with a parameter that depends on time.
The extension to high-temperature dynamics was achieved with the help of a
\emph{space-time cluster expansion} developed in Maes and Netocn\'y~\cite{MaNe02}.
The basic picture that emerged from this work was the following:
\begin{itemize}
\item[(1)]
$\mu_t$ is Gibbs for small $t$;
\item[(2)]
$\mu_t$ is non-Gibbs for intermediate $t$;
\item[(3)]
in zero magnetic field $\mu_t$ remains non-Gibbs for large $t$, while in non-zero
magnetic field $\mu_t$ becomes Gibbs again for large $t$.
\end{itemize}
Further research went into several directions and, roughly summarized, gave the
following results:
\begin{itemize}
\item[(a)]
Small-time conservation of Gibbsianness is robust: this holds for a large class of spin systems and of dynamics, including discrete spins (Le Ny and Redig~\cite{LNRe02}), continuous
spins (Dereudre and Roelly~\cite{DeRo05}, van Enter, K\"ulske, Opoku and Ruszel,
~\cite{KuOp08a}, ~\cite{vEnRu08}, \cite{vEnRu10}, ~\cite{Opthesis09}, ~\cite{vEnKuOpRu}), which can be subjected to Glauber dynamics,
mixed Glauber/Kawasaki dynamics, and interacting-diffusion dynamics,
not even necessarily Markovian (Redig, Roelly and Ruszel~\cite{ReRoRu10}),
appliedto a large class of initial measures (e.g. Gibbs measures
for a finite-range or an exponentially decaying interaction potential).
\item[(b)]
Gibbs-non-Gibbs transitions can also be defined naturally for mean-field models
(see e.g. K\"ulske and Le Ny~\cite{KuLN07} for Curie-Weiss models subject to an
independent spin-flip dynamics). In this context, much more explicit results can
be obtained: transitions are sharp (i.e., in zero magnetic field there is a single
time after which the measure becomes non-Gibbs and stays non-Gibbs forever, and
in non-zero magnetic field there is a single time at which it becomes Gibbs again).
Bad configurations can be characterized explicitly (with the interesting effect
that non-neutral bad configurations can arise below a certain critical temperature). For further developments on mean-field results see also ~\cite{KuOp08b},
~\cite{ErKupr}.
\item[(c)]
Gibbs-non-Gibbs transitions can also occur for continuous unbounded spins subject to
independent Ornstein-Uhlenbeck processes (K\"ulske and Redig~\cite{KuRe06}),
and for continuous bounded spins subject to independent diffusions (Van Enter
and Ruszel~\cite{vEnRu08}, \cite{vEnRu10}), even in two dimensions where no
static phase transitions occur.
\end{itemize}
Bad configurations can be detected by looking at a so-called \emph{two-layer
system:} the joint distribution of the configuration at time $t=0$ and time $t>0$.
If we condition on a particular configuration $\eta$ at time $t>0$, then the
distribution at time $t=0$ is a Gibbs measure with an $\eta$-dependent Hamiltonian
$H^\eta$, which is a \emph{random-field} modification of the original Hamiltonian
$H$ of the starting measure. If, for some $\eta$, $H^\eta$ has a phase transition,
then this $\eta$ is a bad configuration (see Fern\'andez and Pfister~\cite{FePf97}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nature versus nurture}
\label{S1.2}
While these results led to a reasonably encompassing picture, we were unsatisfied
with the strategy of the proofs for the following reason. All proofs rely on two
fortunate facts: (1) the evolutions can be described in terms of space-time interactions;
(2) these interactions correspond to well-studied models in equilibrium statistical
mechanics. In particular, although the most delicate part of the analysis -- the
proof of the onset of non-Gibbsianness -- was accomplished by adapting arguments
developed in previous studies on renormalization transformations, the actual intuition
that led to these results relied on entirely different arguments, based on the behavior
of \emph{conditioned trajectories}. These intuitive arguments, already stated without proof in
our original work~\cite{vEnFedHoRe02}, can be summarized as follows:
\begin{itemize}
\item[(I)]
If a configuration $\eta$ is \emph{good} at time $t$ (i.e., is a point of continuity
of the single-site conditional probabilities), then the trajectory that leads to
$\eta$ is unique, in the sense that there is a single distribution at time $t=0$
that leads to $\eta$ at time $t>0$. In particular, if $t$ is small, then this
trajectory stays close to $\eta$ during the whole time interval $[0,t]$.
\item[(II)]
If a configuration $\eta$ is \emph{bad} at time $t$ (i.e., is a point of essential
discontinuity of the single-site conditional probabilities), then there are at least
two trajectories compatible with the occurrence of $\eta$ at time $t$. Moreover,
these trajectories can be selected by modifying the bad configuration $\eta$
arbitrarily far away from the origin.
\item[(III)]
Trajectories ending at a configuration $\eta$ at time $t$ are the result of a competition
between two mechanisms:
\begin{itemize}
\item
\emph{Nature:} The initial configuration is close to $\eta$, which is not necessarily
typical for the initial measure, and is preserved by the dynamics up to time $t$.
\item
\emph{Nurture:} The initial configuration is typical for the initial measure and the
system builds $\eta$ in a short interval prior to time $t$.
\end{itemize}
\end{itemize}
As an illustration, let us consider the low-temperature zero-field Ising model subject
to an independent spin-flip dynamics. In~\cite{vEnFedHoRe02} we proved that the fully
alternating configuration becomes and stays bad for large $t$. This fact can be understood
according to the preceding paradigm in the following way. Short times do not give the
system occasion to perform a large number of spin-flips. Hence, the most probable way
to see the alternating configuration at small time $t$ is when the system started in
a zero-magnetization-like state and the evolution kept the magnetization zero up to
time $t$. \emph{This is the nature-scenario!} For larger times $t$, a less costly
alternative is to start in a less atypical manner, and to arrive at the alternating
configuration following a trajectory that stays close for as long as possible to the
unconditioned dynamical relaxation. \emph{This is the nurture-scenario!} In this
situation, we can start either from a plus-like state or a minus-like state, as the
difference in probabilistic cost between these two initial states is exponential in
the size of boundary, and thus is negligible with respect to the volume cost imposed
by a constrained dynamics. It is then possible to select between the plus-like and the
minus-like trajectories by picking the alternating configuration in a large block,
then picking either the all-plus or the all-minus configuration outside this block, and
letting the block size tend to infinity.
We see that the above explanation relies on two facts:
\begin{itemize}
\item[(i)]
The existence of a \emph{nature-versus-nurture} transition, as introduced in
\cite{vEnFedHoRe02}.
\item[(ii)]
The existence of several possible trajectories (once the system is in the nurture
regime), all starting from configurations that are typical for the initial measure
(modulo an boundary-exponential cost). These trajectories evolve to the required bad
configuration over a short interval prior to time $t$.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Large deviations of trajectories}
\label{S1.3}
The goal of the present paper is to put rigor into the above qualitative suggestions.
We propose two novel aspects:
\begin{itemize}
\item[(1)]
the development of a suitable large deviation theory
for \emph{trajectories}, in order to estimate the costs of the different
dynamical strategies;
\item[(2)]
the use of \emph{empirical measures} instead of \emph{configurations},
in order to express the conditioning at time $t$.
\end{itemize}
For a translation-invariant spin-flip
dynamics and a translation-invariant initial measure, nothing is lost by moving to the
empirical measure because the bad configurations form a translation-invariant set. Instead,
a lot is gained because, as we will show, the trajectory of the empirical measure satisfies
a \emph{large deviation principle} under quite general conditions on the spin-flip rates
(e.g.\ there is no restriction to high temperature). Moreover, the question of uniqueness
versus non-uniqueness of optimal trajectories (i.e., minimizers of the large deviation
rate function) can be posed and tackled for a large class of dynamics, which places the
dynamical Gibbs-non-Gibbs-transition into a framework where it gains more physical
relevance.
Here is a list of the results presented in the sequel.
\begin{itemize}
\item[(A)]
\emph{Existence of a large deviation principle for trajectories.}
We apply the theory developed in Feng and Kurtz~\cite{FeKu06}, Section 8.6. The rate
function is the integral of the Legendre transform of the generator of the non-linear
semigroup defined by the dynamics. In suitably abstract terms, this generator can be
associated to a Hamiltonian, and the rate function to the integral of a Lagrangian
(Sections~\ref{S2}--\ref{S5}).
\item[(B)]
\emph{Explicit expression for the generator of the non-linear semigroup of the dynamics.}
These are obtained in Theorems~\ref{elprop}--\ref{elpropalt} below (Section~\ref{S3}).
\item[(C)]
\emph{Rate functions for trajectories and associated optimal trajectories.}
The general Legendre-transform prescription is explicitly worked out for a couple of
simple examples, and optimal trajectories are exhibited (Sections~\ref{S4.2}--\ref{S4.3}).
\item[(D)]
\emph{Relation with thermodynamic potentials.}
Relations are shown between the non-linear generator and the derivative of a ``constrained
pressure''. Similarly, the rate function per unit time is related to the Legendre transform
of this pressure (Section~\ref{S5.2}).
\item[(E)]
\emph{Definition of bad measures.}
This definition, introduced in Section~\ref{S6}, is the transcription to our more general
framework of the notion of \emph{bad configuration} used in our original work~\cite{vEnFedHoRe02}.
In Section~\ref{S7} we discuss the possible relations between these two notions of badness.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Future research program}
\label{S1.4}
The results in (A)--(E) above are the preliminary steps towards a comprehensive theory of
dynamical Gibbs-non-Gibbs transitions based on the principles outlined above. Let us conclude
this introduction with a list of further issues which must be addressed to develop such a theory:
\begin{itemize}
\item[$\bullet$]
\emph{Definition of ``nature-trajectories'' and ``nurture-trajectories''.}
This is a delicate issue that requires full exploitation of the properties of the rate
function for the trajectory. It must involve a suitable notion of distance between conditioned
and unconditioned trajectories.
\item[$\bullet$]
\emph{Relation between nature-trajectories and Gibbsianness.}
It is intuitively clear that
\newline
Gibbsianness is conserved for times so short that only
nature-trajectories are possible. A rigorous proof of this fact would confirm our intuition and
would lead to alternative and less technical proofs of short-time Gibbsianness preservation.
\item[$\bullet$]
\emph{Study of nurture-trajectories.}
We expect that nurture-trajectories start very close to unconstrained trajectories, and move
away only shortly before the end in order to satisfy the conditioning. For the case of time-reversible
evolutions, the time it takes to get to the nurture-regime should be the same as the initial
relaxation time to (almost) equilibrium.
\item[$\bullet$]
\emph{Study of nature-nurture transitions.}
Transitions from nature to nurture should happen only once for every conditioning measure (i.e.,
there should be no nature-restoration). Natural questions are: Does the time at which these
transitions take place depend on the conditioning measure? Is there a common time after which
every trajectory becomes nurture?
\item[$\bullet$]
\emph{Case studies of trajectories leading to non-Gibbsianness.}
These should determine ``forbidden regions'' in trajectory space. Natural questions are: How
do these regions evolve? Are they monotone in time?
\item[$\bullet$]
\emph{Relation between nurture-trajectories and non-Gibbsianness.}
While we expect that ``all trajectories are nature'' implies Gibbsianness of the evolved measure,
we do not expect that``some trajectories are nurture'' leads to non-Gibbsianness. Examples are
needed to clarify this asymmetry. The case of the Ising model in non-zero field -- in which
Gibbsianness is eventually restored -- should be particularly enlightening.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Outline}
\label{S1.5}
Our paper is organized as follows. In Section~\ref{S2}, we consider the case of
independent spin-flips, as a warm-up for the rest of the paper. In Section~\ref{S3},
we compute the \emph{non-linear generator} for dependent spin-flips, which plays
a key rol in the large deviation principle we are after. In Sections~\ref{S4}
and \ref{S5}, we compute the \emph{Legendre transform} of this non-linear
generator, which is the object that enters into the associated rate function,
as an \emph{action integral}. In Section~\ref{S4} we do the computation for independent
spin-flips, in Section~\ref{S5} we extend the computation to dependent spin-flips.
In Section~\ref{S6}, we look at \emph{bad measures}, i.e., measures at time $t>0$
for which the optimal trajectory leading to this measure and minimizing the rate
function is non-unique. In Section~\ref{S7}, we use these results to develop our
large-devation view on Gibbs-non-Gibbs transtions. In Appendix~\ref{App} we
illustrate the large deviation formalism in Feng and Kurtz~\cite{FeKu06}, which
lies at the basis of Sections~\ref{S2}--\ref{S5}, by considering a simple example,
namely, a Poisson random walk with small increments. This will help the reader not
familiar with this formalism to grasp the main ideas.
%%%%%%%%%%%%%%%%%% SECTION 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Independent spin-flips: trajectory of the magnetization}
\label{S2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Large deviation principle}
\label{S2.1}
As a warm-up, we consider the example of Ising spins on the one-dimensional torus
$T_N=\{1,\ldots,N\}$ subject to a rate-1 independent spin-flip dynamics. Write
$\pee_N$ to denote the law of this process. We look at the trajectory of the magnetization,
i.e., $t \mapsto m_N(t) = N^{-1}\sum_{i=1}^N \si_i(t)$, where $\si_i(t)$ is the spin
at site $i$ at time $t$. A spin-flip from $+1$ to $-1$ (or from $-1$ to $+1$) corresponds
to a jump of size $-2N^{-1}$ (or $+2N^{-1}$) in the magnetization, i.e., the generator
$L_N$ of the process $(m_N(t))_{t \geq 0}$ is given by
\be{genma}
(L_N f)(m) = \tfrac{1+m}{2}\,N\left[f\left(m-2N^{-1}\right)-f(m)\right]
+\tfrac{1-m}{2}\,N\left[f\left(m+2N^{-1}\right)-f(m)\right]
\ee
for $m\in\{-1,-1+2N^{-1},\ldots,1-2N^{-1},1\}$. If $\lim_{N\to\infty} m_N=m$ and
$f$ is $C^1$ with bounded derivative, then
\be{LNlim}
\lim_{N\to\infty} (L_N f)(m_N) = (Lf)(m) \quad \mbox{ with } \quad (Lf)(m) = -2mf'(m).
\ee
This is the generator of the deterministic process $m(t) = m(0)e^{-2t}$, solving
the equation $\dot{m}(t) = -2m(t)$ (the dot denotes the derivative with respect
to time).
The trajectory of the magnetization satisfies a large deviation principle, i.e.,
for every time horizon $T\in (0,\infty)$ and trajectory $\gamma=(\gamma_t)_{t\in[0,T]}$,
\be{infolarge}
\pee_N\Big(\big(m_N(t)\big)_{t \in [0,T]} \approx (\gamma_t)_{t \in [0,T]}\Big)
\approx \exp\left[- N \int_0^T L(\gamma_t,\dgam_t)\,dt\right],
\ee
where the Lagrangian $t \mapsto L(\gamma_t,\dgam_t)$ can be computed following the
scheme of Feng and Kurtz~\cite{FeKu06}, Example 1.5. Indeed, we first compute the
so-called \emph{non-linear generator} $H$ given by
\be{nonlingen}
(Hf)(m) = \lim_{N\to\infty} (\caH_N f)(m_N)
\quad \mbox{ with } \quad (\caH_N f)(m_N)
= \frac{1}{N}\,e^{-N f(m_N)}\,L_N (e^{N f})(m_N),
\ee
where $\lim_{N\to\infty} m_N=m$. This gives
\be{nonlingen1}
(Hf)(m) = \tfrac{m+1}{2}\,(e^{-2f'(m)}-1) + \tfrac{1-m}{2}\,(e^{2f'(m)}-1),
\ee
which is of the form
\be{Hform}
(Hf)(m) = H\big(m,f'(m)\big)
\ee
with
\be{hamiltonian}
H(m,p) = \tfrac{m+1}{2}\,(e^{-2p}-1) + \tfrac{1-m}{2}\,(e^{2p}-1).
\ee
Because $p \mapsto H(m,p)$ is convex, we have
\be{legen}
H(m,p) = \sup_{q\in\R}\,[pq-L(m,q)]
\ee
with
\be{kul}
\begin{aligned}
L(m,q) &= \sup_{p\in\R}\,[pq - H(m,p)]\\
&= \frac{q}{2} \log\left(\frac{q+\sqrt{q^2+4(1-m^2)}}{2(1+m)}\right)
-\frac12\sqrt{q^2+4(1-m^2)}+1.
\end{aligned}
\ee
Hence, using the theory developed in Feng and Kurtz~\cite{FeKu06}, Chapter 1,
Example 1.5, we indeed have the large deviation principle in \eqref{infolarge}
with $L(\gamma_t,\dgam_t)$ given by \eqref{kul} with $m=\gamma_t$ and $q=\dgam_t$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Optimal trajectories}
\label{S2.2}
We may think of the typical trajectories $(m_N(t))_{t\in[0,T]}$ as being exponentially
close to \emph{optimal trajectories} minimizing the \emph{action functional} $\gamma
= (\gamma_t)_{t \in [0,T]} \mapsto \int_0^T L(\gamma_t,\dgam_t)\,dt$. The optimal
trajectories satisfy the Euler-Lagrange equations
\be{Euler}
\frac{d}{dt}\,\frac{\partial L}{\partial\dgam_t}
= \frac{\partial L}{\partial \gamma_t}
\ee
or, equivalently, the Hamilton-Jacobi equations corresponding to the Hamiltonian
in \eqref{hamiltonian},
\be{hameq}
\dot{m} = \frac{\partial H}{\partial p}, \qquad
\dot{p} = -\frac{\partial H}{\partial m},
\ee
which gives
\be{hameq1}
\dot{m} = -m(e^{2p}+e^{-2p})+(e^{2p}-e^{-2p}), \qquad
\dot{p} = \tfrac12\,(e^{2p}- e^{-2p}).
\ee
Putting $h= \tanh(p)$ and integrating the second equation in \eqref{hameq1}, we obtain
\be{hameq2}
h(t)= C\,e^{2t}.
\ee
Using that $\mathrm{arctanh}(x)=\tfrac12\log(\tfrac{1+x}{1-x})$, we get
\be{hameq3}
\dot{m} = -m\,\frac{2+2h^2}{1-h^2}+\frac{4h}{1-h^2},
\ee
which can be integrated to yield the solution
\be{mGopt}
m(t) = C_1 e^{2t} + C_2 e^{-2t},
\ee
where the constants $C_1,C_2$ are determined by the initial magnetization
and the corresponding initial momentum. One example of an optimal trajectory
corresponds to the dynamics \emph{starting} from an initial magnetization
$m_0$, giving $m(t) = m_0e^{-2t}$, i.e., $C_1=0$ and $C_2=m_0$. Another
example of an optimal trajectory is the reversed dynamics \emph{arriving}
at magnetization $m_T$ at time $T$, giving $m(t) = m_Te^{2(t-T)}$, i.e.,
$C_2=0$ and $C_1= m_Te^{-2T}$.
Yet another example is the following. Suppose that we start the independent
spin-flip dynamics from a measure under which the magnetization satisfies a
large deviation principle with rate function, say, $I$, e.g.\ a Gibbs measure.
If we want to arrive at a given magnetization $m_T$ at time $T$, then the
optimal trajectory is given by \eqref{mGopt} with end condition $m(T)=m_T$
and satisfying the \emph{open-end condition} relating the Lagrangian $L$ at
time $t=0$ to the rate function $I$ at magnetization $m=\gamma_0$ as follows:
\be{freeend}
\left[\frac{\partial L (\gamma_t,\dgam_t)}
{\partial \dot{\gamma_t}}\right]_{t=0}
= - \left[\frac{\partial I(m)}{\partial m}\right]_{m=\gamma_0}.
\ee
This condition is obtained by minimizing $\gamma \mapsto I(\gamma_0) + \int_0^T
L(\gamma_t,\dgam_t)\,dt$ (see Ermolaev and K\"ulske~\cite{ErKupr}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Trajectory of the empirical measure for dependent spin-flips }
\label{S3}
We will generalize the computation in Section~\ref{S2} in two directions.
First, for independent spin-flips we are confronted with the problem that
the rate at which the average of a local observable changes in general
depends on the average of other observables. Second, for dependent
spin-flips even the trajectory of the magnetization is not Markovian.
Therefore, we are obliged to consider the time evolution of all spatial
averages jointly, i.e., the \emph{empirical measure}.
%%%%%%%%%%%%%%%%%%%
\subsection{Setting and notation}
\label{S3.1}
For $N\in\N$, let $\tend$ be the $d$-dimensional $N$-torus $(\Z/(2N+1)\Z)^d$.
For $i,j\in\tend$, let $i+j$ denote coordinate-wise addition modulo $2N+1$. We
consider Glauber dynamics of Ising spins located at the sites of $\tend$, i.e.,
on the configuration space $\Omega_N = \{-1,1\}^{\tend}$. We write $\Omega =
\{-1,1\}^{\Zd}$ to denote the infinite-volume configuration space. Configurations
are denoted by symbols like $\si$ and $\eta$. For $\si\in\Omega_N$, $\si_i$
denotes the value of the spin at site $i$. We write $\cM_1(\Omega)$ to denote
the set of probability measures on $\Omega$, and similarly for $\cM_1(\Omega_N)$.
The dynamics is defined via the generator $L_N$ acting on functions $f\colon\,
\Omega_N\to\R$ as
\be{geno}
(L_N f)(\si) = \sum_{i\in\tend} c_i(\si)\,[f(\si^i)-f(\si)],
\ee
where $\si^i$ denotes the configuration obtained from $\si$ by flipping the spin
at site $i$. The rates $c_i(\si)$ are assumed to be strictly positive and
translation invariant, i.e.,
\be{transrate}
c_i (\si) = c_0 (\tau_i \si) = c(\tau_i \si) \quad \mbox{ with }
\quad (\tau_i\si)_j = \si_{i+j}.
\ee
We think of the dynamics with generator $L_N$ as a finite-volume version with
periodic boundary condition of the infinite-volume generator
\be{infvol}
(Lf)(\si) = \sum_{i\in\Zd} c_i(\si)\,[f(\si^i)-f(\si)],
\ee
where now $f$ is supposed to be a local function, i.e., a function depending on a finite
number of $\si_j$, $j\in\Zd$. We denote by $(S_t)_{t \geq 0}$ with $S_t = e^{tL}$
the semigroup acting on $C(\Omega)$ (the space of continuous functions on $\Omega)$)
associated with the generator in \eqref{infvol}, and similarly $(S^N_t)_{t \geq 0}$
with $S^N_t= e^{tL_N}$. For $\mu\in\cM_1(\Omega)$, we denote by $\mu S_t\in\cM_1
(\Omega)$ the distribution $\mu$ evolved over time $t$, and similarly for $\mu_N
S^N_t$ and $\mu_N\in\cM_1(\Omega_N)$.
We embed $\tend$ in $\Z^d$ by identifying it with $\Lambda_N^d=([-N,N]\cap\Z)^d$.
Through this identification, we give meaning to expressions like $\sum_{i\in\tend}
f(\tau_i \si)$ for $\si\in\Omega$ and $f\colon\Omega\to\R$. In this way we may also
view local functions $f\colon\Omega\to\R$ as functions on $\Omega_N$ as soon as $N$
is large enough for $\Lambda_N^d$ to contain the dependence set of $f$. For a
translation-invariant $\mu\in\cM_1(\Omega)$, we denote by $\mu_N$ its natural
restriction to $\Omega_N$.
By the locality of the spin-flip rates, the infinite-volume dynamics is well-defined
and is the uniform limit of the finite-volume dynamics, i.e., for every local function
$f\colon\,\Omega\to\R$ and $t>0$,
\be{}
\lim_{N\to\infty} \|S_t^N f-S_tf\|_\infty =0.
\ee
See Liggett~\cite{Li85}, Chapters 1 and 3, for details on existence of the
infinite-volume dynamics.
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Empirical measure}
\label{S3.2}
For $N\in\N$ and $\si\in\Omega_N$, the empirical measure associated with $\si$ is
defined as
\be{empir}
\loc_N (\si) = \frac{1}{|\tend|} \sum_{i\in\tend} \delta_{\tau_i\si}.
\ee
This is an element of $\cM_1(\Omega_N)$ which acts on functions $f\colon\,\Omega_n
\to\R$ as
\be{}
\langle f,\loc_N\rangle = \int_{\Omega_N} f\,d\loc_N
= \frac1\norm\sum_{i\in\tend} f(\tau_i \si).
\ee
As already mentioned above, a local $f\colon\Omega\to\R$ may be considered as a
function on $\Omega_N$ for $N$ large enough. A sequence $(\mu_N)_{N\in\N}$
with $\mu_N\in\cM_1(\Omega_N)$ converges weakly to some $\mu\in\cM_1(\Omega)$
if
\be{}
\lim_{N\to\infty} \int_{\Omega_N} fd\mu_N = \int_\Omega fd\mu
\qquad \forall\,f \mbox{ local}.
\ee
For $\si\in\Omega$, we define its periodized version $\si^N$ as $\si^N_i= \si_i$
for $i= (i_1,\ldots,i_d)$ with $-N\leq i_k0$, $i=1,\ldots,n$. Put $C_\mu=\sum_{i=1}^n c_i\mu_i$. In the calculation with
$c_i=1$, $i=1,\ldots,n$, this ``total mass'' does not depend on $\mu$ and is equal to 1.
Now, however, it becomes a normalization that depends on $\mu$. We say that $(D^T)^{-1}
(\alpha,\mu)$ is well-defined if there exists a non-negative vector $\nu=\nu(\alpha,\mu)
=(\nu_1\ldots,\nu_n)$ with sum $C_\mu$ such that $D^T\nu = \alpha$. The analogue of
\eqref{lagrafindim} reads
\be{lagrafindimc}
L(\mu,\alpha) = \left\langle\log\left[\frac{(D^T)^{-1}(\alpha,\mu)}{\mu}\right],
(D^T)^{-1}(\alpha,\mu)\right\rangle.
\ee
In order to find the analogue of this expression in the infinite-dimensional setting,
we proceed as follows. For two finite positive measures $\mu,\nu$ of equal total
mass $M$, we define $S(\mu|\nu)$ to be the relative entropy density of the probability
measures $\mu/M, \nu/M$, i.e., $S(\mu|\nu)=s(\nu/M|\mu/M)$. For $\mu\in\cM_1(\Omega)$,
we define the $c$-modification of $\mu$ as the positive measure defined via $\int_\Omega
f(\si)d\mu_c(\si)=\int_\Omega c(\si)f(\si)d\mu(\si)$. For a signed measure of total mass
zero and $\mu\in\cM_1(\Omega)$, we say that $(\caD^*)^{-1}(\alpha,\mu,c)=\nu$ is
well-defined if there exists a positive measure $\nu$ of total mass equal to that of
$\mu_c$ such that $\caD^*(\nu)=\alpha$. Then the analogue of \eqref{lagrafindimc} becomes
\be{lagrainfdimc}
L(\mu,\alpha)= s\left((\caD^*)^{-1}(\alpha,\mu,c)|\mu_c\right).
\ee
%%%%%%%%%%%%%%%%%%%%%%
\subsection{The non-linear semigroup and its relation with relative entropy}
\label{S5.2}
The non-linear semigroup with generator \eqref{nonlin} is defined as follows. Let
$\caP^{\mathrm{inv}}(\Omega)$ be the set of translation-invariant probability measures
on $\Omega$. For local functions $f_1,\ldots,f_n\colon\,\Omega\to\R$ and a
$\ce^\infty$-function $\Psi\colon\,\R^n\to\R$, we define an associated function
$F^{f_1,\ldots,f_N}_\Psi\colon\,\caP^{\mathrm{inv}}(\Omega)\to\R$ via
\be{}
F^{f_1,\ldots,f_n}_\Psi (\mu)
= \Psi\left(\int_\Omega f_1 d\mu,\ldots,\int_\Omega f_n d\mu\right).
\ee
Since $\langle f_i,\loc_N\rangle$ is well-defined for $N$ large enough, we can
define $F^{f_1,\ldots,f_n}_\Psi (\loc_N)$ for $N$ large enough as well. This allows
us to define a non-linear semigroup $(V(t))_{t\geq 0}$ via
\be{semigrogi}
\big(V(t) F^{f_1,\ldots,f_n}_\Psi\big)(\mu) = \lim_{N\to\infty} \frac{1}{\norm}\,
\log\E_{\sigma^N}\left(\exp\left[\norm F^{f_1,\ldots,f_n}_\Psi\big(\loc_N(\si^N(t))\big)
\right]\right),
\ee
where $\E_{\sigma^N}$ denotes expectation with respect to the law of the process
starting from $\sigma^N$, and the limit is taken along a sequence of configurations
$(\si^N)_{N\in\N}$ with $\si^N\in\Omega_N$ such that the associated empirical measure
$\loc_N(\si^N)$ converges weakly to $\mu$ as $N\to\infty$. If $V(t)$ exists, then it
defines a non-linear semigroup, and the generator of $V(t)$ is given by
\eqref{nonlinsemgen}.
Conversely, if $H$ in \eqref{nonlinsemgen} generates a semigroup, then this must be
$(V(t))_{t\geq 0}$. The fact that this semigroup is well-defined is sufficient
to imply the large deviation principle for the trajectory of the empirical measure
(Feng and Kurtz~\cite{FeKu06}, Theorem 5.15). Technically, the difficulty consists
in showing that the generator in \eqref{nonlinsemgen} actually generates a semigroup.
We now make the link between the non-linear semigroup, its generator and some
familiar objects of statistical mechanics, such as pressure and relative entropy
density.
\bd{cpres}
The constrained pressure at time $t$ associated with a function $f\colon\,\Omega\to\R$
and a Gibbs measure $\mu\in\caP^{\mathrm{inv}}(\Omega)$ is defined as
\be{pres}
p_t(f|\mu) = \lim_{N\to\infty} \frac{1}{\norm}\,
\log\E_{\si^N}\left(e^{\sum_{x\in \T_N} \tau_x f(\si_t)}\right),
\ee
where the limit is taken along a sequence of configurations $(\si^N)_{N\in\N}$
with $\si^N\in\Omega_N$ such that the associated empirical measure $\loc_N(\si^N)$
converges weakly to $\mu$ as $N\to\infty$.
\ed
In particular, $p_0 (f|\mu)= \int_\Omega f d\mu$. The relation between the non-linear
semigroup in \eqref{semigrogi} and the constrained pressure at time $t$ reads
\be{pressem}
\big(V(t)\langle f,\cdot\rangle\big)(\mu) = p_t(f|\mu).
\ee
The pressure at time $t$ is defined as
\be{presst}
p(f|\mu_t) = \lim_{N\to\infty} \frac{1}{\norm}\,
\log\E_{\mu}\left(e^{\sum_{x\in \T_N} \tau_x f(\si_t)}\right).
\ee
This is well-defined as soon as the dynamics starts from a Gibbs measure $\mu_0=\mu$ (see
Le Ny and Redig~\cite{LNRe04}). The relation between the pressure and the constrained
pressure reads
\be{pressrel}
p(f|\mu_t) = \sup_{\nu\in\caP^{\mathrm{inv}}(\Omega)}\,[p_t(f|\nu) -s(\nu|\mu)].
\ee
On the other hand, the pressure at time $t$ is the Legendre transform of the relative
entropy density with respect to $\mu_t$, i.e.,
\be{legentrop}
p(f|\mu_t) = \sup_{\nu\in\caP^{\mathrm{inv}}(\Omega)}\,
\left[\int_\Omega f d\nu- s(\nu|\mu_t)\right],
\ee
where the relative entropy density $s(\nu|\mu_t)$ exists because $\mu_t$ is
\emph{asymptotically decoupled} (see Pfister~\cite{Pf??}) as soon as
$\mu_0=\mu$ is a Gibbs measure (see Le Ny and Redig~\cite{LNRe04}).
The relation between the non-linear generator and the constrained pressure is now
as follows. Define the Legendre transform of the constrained pressure as
\be{legendrepress}
p^*_t (\nu|\mu) = \sup_{f\in\C(\Omega)} \left[\int f d\nu - p_t (f|\mu)\right].
\ee
Then the relation with the Hamiltonian in \eqref{hamilton} and the Lagrangian in
\eqref{lagrainfdimc} is
\be{hampress}
H(\mu,f) = \left[\frac{d}{dt}\,p_t(f|\mu)\right]_{t=0}
\ee
and
\be{lagpress}
L(\mu,\alpha) = \lim_{t\to 0} \frac{1}{t}\,p^*_t(\mu+t\alpha|\mu).
\ee
\medskip\noindent
{\bf Remark:} The operator $\caD$, acting on the space $C(\Omega)$ of continuous functions
on $\Omega$, has a dual operator $\caD^*$, acting on the space $\cM(\Omega)$ of finite signed
measures on $\Omega$, defined via
\be{}
\langle f,\caD^*\mu\rangle = \langle\caD f,\mu\rangle.
\ee
In order to gain some understanding for $\caD^*$ (which will be useful later on),
we first compute $\caD^*$ for a Gibbs measure $\mu\in\caP^{\mathrm{inv}}(\Omega)$.
Without loss of generality we may assume that the interaction potential of $\mu$
is a sum of terms of the form $\Phi(A,\si)=J_A H(A,\si)$, $A\subset\Zd$ finite, where $J_A$ is
translation invariant, i.e., $J_{A+k} = J_A$, $k\in\Zd$. We also assume absolute
summability, i.e.,
\be{}
\sum_{A\ni 0} |J_{A}| <\infty.
\ee
Remember that
\be{}
(\caD f)(\si) = \sum_{j\in\Zd} \left[f(\tau_j(\si^0))- f(\tau_j(\si))\right].
\ee
Therefore, for the Gibbs measure $\mu$ under consideration, we have
\be{gibbsd}
\langle\mu,\caD f\rangle = \int_\Omega\left(\sum_{j\in\Zd}\frac{d\mu^0}{d\mu}
\circ\tau_{-j}-1\right)\,fd\mu,
\ee
where $\mu^0$ denotes the distribution of $\si^0$ when $\si$ is distributed according
to $\mu$. Note that the sum in the right-hand side of \eqref{gibbsd} is formal, i.e.,
the integral is well-defined due to the multiplication with the local function $f$.
In terms of $J_A$, $A\subset\Zd$ finite, we have
\be{}
\left(\sum_{j\in\Zd} \frac{d\mu^0}{d\mu}\circ\tau_{-j}-1\right)(\si)
= \sum_{j\in\Zd} \left(e^{-\sum_{A\ni 0}-2J_A H(A-j,\si)}-1\right),
\ee
where, once again, this expression is well-defined only after multiplication with a
local function and integrated over $\mu$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bad empirical measures}
\label{S6}
In Section \ref{S7} we will see what consequences the large deviation principle for
the trajectory of the empirical measure derived in Sections \ref{S3} and \ref{S5}
has for the question of Gibbs versus non-Gibbs. This needs the notion of bad
empirical measure, which we define next.
If we start our spin-flip dynamics from a Gibbs measure $\mu\in\caP^{\mathrm{inv}}
(\Omega)$, then a probability-measure-valued trajectory $\gamma=(\gamma_t)_{t\in[0,T]}$
has cost
\be{}
\caI_\mu (\gamma) = s(\gamma_0|\mu) + \int_0^T L(\gamma_t,\dot{\gamma}_t)\,dt,
\ee
where the term $s(\gamma_0|\mu)$ is the cost of the initial distribution $\gamma_0$.
We are interested in the minimizers of $\caI_\mu (\gamma)$ over the set of trajectories
$\gamma$ that {\em end} at a given measure $\nu$. Let
\be{}
\begin{aligned}
K_T (\mu',\nu) &= \inf_{\gamma\colon\,\gamma_0 = \mu',\gamma_T=\nu}
\int_0^T L(\gamma_t,\dot{\gamma}_t)\,dt\\
&= - \lim_{N\to\infty} \frac{1}{\norm}\,
\log\pee_\mu\left(\loc_N(\si_T)=\nu|\loc_N(\si_0)=\mu'\right).
\end{aligned}
\ee
Then $e^{\norm -K_T(\mu',\nu)}$ can be thought of as the transition probability
for the empirical measure $\loc_N$ to go from $\mu'$ to $\nu$, up to factors of order
$e^{o(\norm)}$. Hence
\be{sK}
\begin{aligned}
&-\frac{1}{\norm}\,\log\pee_\mu\left(\loc_N(\si_0)= \mu'|\loc_N (\si_T)=\nu\right)\\
&\qquad = [s(\mu'|\mu) + K_T (\mu',\nu)]
- \inf_{\mu'\in\caP^{\mathrm{inv}}(\Omega)}\,[s(\mu'|\mu) + K_T (\mu',\nu)].
\end{aligned}
\ee
Let $M^*(\mu,\nu)$ be the set of probability measure $\mu'$ for which the infimum
in the right-hand side of \eqref{sK} is attained. We can then think of each element
in this set as a typical empirical measure at time $t=0$ given that the empirical
measure at time $T$ is $\nu$. When $M^*$ is a singleton, we denote its unique element
by $\mu^*(\mu,\nu)$.
\bd{bad}
(a) A measure $\nu$ is called bad at time $t$ if $M^*(\mu,\nu)$ contains at least
two elements $\mu_1$ and $\mu_2$ and there exist two sequences $(\nu^1_n)_{n\in\N}$
and $(\nu^2_n)_{n\in\N}$, both converging to $\nu$ as $n\to\infty$, such that
$\mu^*(\mu,\nu^1_n)$ converges to $\mu_1$ and $\mu^*(\mu,\nu^2_n)$ converges
to $\mu_2$.\\
(b) A measure $\nu$ that is bad at time $t$ has at least two possible histories,
stated as a two-layer property: seeing the measure $\nu$ at time $t$ is compatible
(in the sense of optimal trajectories) with two different measures at time $t=0$.
\ed
\noindent
Badness of a measure can be detected as follows.
\bp{badprop}
A measure $\nu$ is bad at time $t$ if there exists a local function $f\colon\Omega\to\R$,
two sequences $(\nu^1_n)_{n\in\N}$ and $(\nu^2_n)_{n\in\N}$ both converging to
$\nu$, and an $\epsi>0$ such that
\be{baddefi}
\Big|\E\Big(f(\si(0))~|~\loc_N(\si(t))=(\nu^1_n)_N\Big)
-\E\Big(f(\si(0))~|~\loc_N(\si(t))=(\nu^2_n)_N\Big)\Big| > \epsi \quad \forall\,N,n\in\N,
\ee
where $(\nu_n)_N$ denotes the projection of $\nu_n$ on $\tend$.
\ep
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A large deviation view on dynamical Gibbs-non-Gibbs transitions}
\label{S7}
In van Enter, Fern\'andez, den Hollander and Redig~\cite{vEnFedHoRe02} we studied
the evolution of a Gibbs measure $\mu$ under a high-temperature spin-flip dynamics.
We showed that the Gibbsianness of the measure $\mu_t$ at time $t>0$ is equivalent
to the absence of a phase transition in the double-layer system. More precisely,
conditioned on end configuration $\eta$ at time $t$, the distribution at time $t=0$
is a Gibbs measure $\mu^\eta$ with $\eta$-dependent formal Hamiltonian
\be{}
H^\eta_t (\si, \eta) = H(\si) + h_t \sum_{i\in\Zd} \si_i \eta_i,
\ee
where $t\mapsto h_t$ is a monotone function with $\lim_{t\downarrow 0} h_t=\infty$
and $\lim_{t\to\infty} h_t=0$. If the double-layer system has a phase transition for
an end configuration $\eta$, then $\eta$ is called bad. In that case $\eta$ is an
essential point of discontinuity for any version of the conditional probability
$\mu_t (\si_\la=\cdot\,|\si_{\la^c})$, $\la\subset\Zd$ finite.
The relation between the double-layer system and the trajectory of the empirical
distribution is as follows. Suppose that the double-layer system has no phase
transition for any end configuration $\eta$. If we condition the empirical
measure at time $t>0$ to be $\nu$, then (by further conditioning on the configuration
$\eta$ at time $t>0$) we conclude that at time $t=0$ we have the measure $\int_\Omega
\mu^\eta\nu(d\eta)$. Hence the optimizing trajectory is unique. Conversely, if there exist
a bad configuration $\eta$, then (because of the translation invariance of the
initial measure and of the dynamics) all translates of $\eta$ are bad also. Hence
we expect that a translation-invariant measure $\nu$ arising as any weak limit
point of $\norm^{-1}\sum_{x\in\tend}\delta_{\tau_x\eta}$ is bad also.
As an example, let us consider a situation studied in \cite{vEnFedHoRe02}. The
dynamics starts from $\mu^{+}_\beta$, the low-temperature plus-phase of the Ising
model with zero magnetic field, and evolves according to independent spin-flips.
Then, from some time onwards, the alternating configuration $\eta_{\mathrm{alt}}
(x)=(-1)^{\sum_{i=1}^d |x_i|}$ becomes bad. The same is true for $-\eta_{\mathrm{alt}}$,
and so the translation-invariant measure
\be{nusplit}
\nu = \tfrac12\left(\delta_{\eta_{\mathrm{alt}}}+\delta_{-\eta_{\mathrm{alt}}}\right)
\ee
has the property that, for $\nu$-a.e.\ configuration $\eta$, the double-layer system
has a phase transition when the end configuration is $\eta$. Moreover, the Hamiltonian
$H^\eta_t$ has a plus-phase $\mu^+_\eta$ and a minus-phase $\mu^-_\eta$. Therefore,
when we condition on the empirical measure in \eqref{nusplit} we get two possible optimal
trajectories, one starting at $\frac12(\mu^+_\eta + \mu^{+}_{-\eta})$ and one starting
at $\frac12(\mu^-_\eta + \mu^{-}_{-\eta})$. To realize the approximating measures of
Proposition~\ref{badprop}, we choose $\nu^1_n,\nu^2_n$ to be the randomized versions of
$\nu$ where we first choose a configuration according to $\nu$ and then
independently flip spins with probability $1/n$,
to change either from minus to plus or stay plus if it was plus to begin with,
respectively to change to minus or stay minus.
Clearly, by the FKG-inequality, when conditioning
on $\nu^1_n$, respectively, $\nu^2_n$ as empirical distribution, we get a measure at
time $t=0$ that is above $\mu^+_\eta + \mu^+_{-\eta}$, respectively, below $\mu^-_\eta+
\mu^-_{-\eta}$. Hence \eqref{baddefi} holds with $f(\si)=\si_0$, and $\nu$ is bad.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{A simple example of the Feng-Kurtz formalism}
\label{App}
\subsection{Poisson walk with small increments}
\label{A1}
In order to introduce the general formalism developed in Feng and Kurtz~\cite{FeKu06},
let us consider a simple example where computations are simple yet the fundamental
objects of the general theory already appear naturally.
Let $X_N=(X_N(t))_{t \geq 0}$ be the continuous-time random walk on $\R$ that jumps
$N^{-1}$ forward at rate $bN$ and $-N^{-1}$ backward at rate $dN$, with $b,d \in
(0,\infty)$. This is the Markov process with generator
\begin{equation}
\label{poissongen}
(L_N f)(x) = bN \left[f\left(x+N^{-1}\right) - f(x)\right]
+ dN \left[f\left(x-N^{-1}\right) - f(x)\right].
\end{equation}
Clearly, if $\lim_{N\to\infty} X_N(0)=x\in\R$, then
\begin{equation}
\lim_{N\to\infty} X_N(t) = x + (b-d)t, \qquad t>0,
\end{equation}
i.e., in the limit as $N\to\infty$ the random process $X_N$ becomes a deterministic
process $(x(t))_{t\geq 0}$ solving the limiting equation
\begin{equation}
\dot{x} =(b-d), \qquad x(0)=x.
\end{equation}
For all $N\in \N$, we have
\begin{equation}
X_N(t) = N^{-1}\,\left[\cN^+(Nbt)-\cN^-(Ndt)\right] = \sum_{i=1}^N (X_i^{bt}-Y_i^{dt})
\end{equation}
with $\cN^+=(\cN^+(t))_{t\geq 0}$ and $\cN^-=(\cN^-(t))_{t\geq 0}$ independent rate-1
Poisson processes, and $X_i^t$, $Y_i^t$, $i=1,\dots,N$, independent Poisson random
variables with mean $bt$, respectively, $dt$. Consequently, we can use Cram\'er's
theorem for sums of i.i.d.\ random variables to compute
\begin{equation}
\label{poissonrate}
I(at) = \lim_{N\to\infty} \frac{1}{N} \log\pee_N
\big(X_N (t) = at \mid X_N(0)=0\big)
= \sup_{\lambda\in\R} \big[at\lambda - F(\lambda)\big],
\end{equation}
where
\begin{equation}
F(\lambda) = \lim_{N\to\infty} \frac{1}{N} \log \E_N\left(e^{\lambda N X_N(t)}\right)
= b \big(e^{\lambda}-1\big) + d \big(e^{-\lambda}-1\big).
\end{equation}
Thus, we see that
\begin{equation}
I(at) = t L(a)
\end{equation}
with
\begin{equation}
\label{poissonlagra}
L(a) = \sup_{\lambda \in\R} \left[a\lambda-b\big(e^{\lambda}-1\big)
- d \big(e^{-\lambda}-1\big)\right].
\end{equation}
Using the property that the increments of the Poisson process are independent over disjoint time intervals,
we can now compute
\begin{equation}
\label{ratecomputation}
\begin{aligned}
&\lim_{N\to\infty} \frac{1}{N}\log\pee_N\Big((X_N (t))_{t\in [0,T]}
\approx (\gamma_t)_{t\in [0,T]}\Big)\\
&\qquad =
\lim_{n\to\infty}
\sum_{i=1}^n \lim_{N\to\infty} \frac{1}{N}\log\pee_N\Big(X_N(t_i)-X_N(t_{i-1})
\approx \dot{\gamma}_{t_{i-1}}(t_i-t_{i-1})\Big)\\
&\qquad = \lim_{n\to\infty} \sum_{i=1}^n (t_i-t_{i-1})\,L\big(\dot{\gamma}_{t_{i-1}}\big)
= \int_0^T L\big(\dot{\gamma}_t\big)\,dt,
\end{aligned}
\end{equation}
where $L$ is given by \eqref{poissonlagra} and $t_i$, $i=1,\ldots,n$, is a partition of the
time interval $[0,T]$ that becomes dense in the limit as $n\to\infty$.
We see from the above elementary computation that, in the limit as $N\to\infty$,
\begin{equation}
\label{deville}
\pee_N\Big((X_N(t))_{t\in[0,T]} \approx (\gamma(t))_{t\in[0,T]}\Big)
\approx \exp\left[-N\int_0^T L(\gamma_t,\dot{\gamma}_t)\,dt\right],
\end{equation}
where the \emph{Lagrangian} $L$ only depends on the second variable, namely,
\begin{equation}
L(\gamma_t,\dot{\gamma}_t)= L(\dot{\gamma}_t)
\end{equation}
with $L$ given by \eqref{poissonlagra}. We interpret \eqref{deville} as follows: if the
trajectory is not differentiable at some time $t \in [0,T]$, then the probability in the
left-hand side of \eqref{deville} decays superexponentially fast with $N$, i.e.,
\begin{equation}
\lim_{N\to\infty} \frac{1}{N}\log\pee_N\Big((X_N(t))_{t \in [0,T]}
\approx (\gamma_t)_{t \in [0,T]}\Big) = -\infty,
\end{equation}
and otherwise it is given by the formula in \eqref{deville} (read in the standard
large-deviation language).
The \emph{Lagrangian} in \eqref{poissonlagra} is the Legendre transform of the \emph{Hamiltonian}
\begin{equation}
H(\lambda) = b\big(e^{\lambda}-1\big) - d \big(e^{-\lambda}-1\big).
\end{equation}
This Hamiltonian can be obtained from the generator in \eqref{poissongen} as follows:
\begin{equation}
H(\lambda) = \lim_{N\to\infty} \frac{1}{N}\,e^{-Nf_\lambda(x)}\,
\big(L_N e^{N f_\lambda}\big)(x), \qquad f_\lambda(x) = \lambda x.
\end{equation}
More generally, by considering the operator
\begin{equation}
\label{poissonnonlingen}
(\caH f)(x) = \lim_{N\to\infty} \frac{1}{N}\,e^{-N f(x)}\,\left(L_N e^{Nf}\right)(x)
= b \big(e^{f'(x)}-1\big) - d \big(e^{-f'(x)}-1\big),
\end{equation}
we see that the Hamiltonian equals
\begin{equation}
H(\lambda) = (\caH f_\lambda)(x),
\end{equation}
and that, by the convexity of $\lambda \mapsto H(\lambda)$,
\begin{equation}
(\caH f)(x) = H(f'(x)) = \sup_{a\in \R} [a f'(x)- L(a)].
\end{equation}
The operator $\caH$ is called the \emph{generator of the non-linear semigroup}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The scheme of Feng and Kurtz}
\label{A2}
The scheme that produces the Lagrangian in \eqref{poissonlagra} from the operator in
\eqref{poissonnonlingen} actually works in much greater generality. Consider a sequence
of Markov processes $X=(X_N)_{N\in\N}$ with $X_N=(X_N(t))_{t\geq 0}$, living on a common
state space (like $\R$, $\R^d$ or a space of probability measures). Suppose that $X_N$
has generator $L_N$ and in the limit as $N\to\infty$ converges to a process $(x(t))_{t\geq 0}$,
which can be either deterministic (as in the previous example) or stochastic. We want to
identify the Lagrangian controlling the large deviations of the trajectories:
\begin{equation}
\label{pathspacelargedevpri}
\pee_N\Big((X_N (t))_{t \in [0,T]} \approx (\gamma_t)_{t \in [0,T]}\Big)
\approx \exp\left[-N \int_0^T L(\gamma_t,\dot{\gamma}_t)\,dt\right].
\end{equation}
Omitting technical conditions, we see that this can be done in four steps:
\begin{enumerate}
\item
Compute the generator of the non-linear semigroup
\begin{equation}
\label{nonlingengen}
(\caH f)(x) = \lim_{N\to\infty} \frac{1}{N}\,e^{-N f(x)}\,\left(L_N e^{N f}\right)(x).
\end{equation}
\item
Look for a function $H(x,p)$ of two variables such that
\begin{equation}
(\caH f)(x) = H(x,\nabla f(x)).
\end{equation}
What $\nabla f$ means depends on the context: on $\R^d$ it simply is the gradient of $f$,
while on an infinite-dimensional state space it is a functional derivative.
\item
Express the function $H$ as a Legendre transform:
\begin{equation}
H(x,p) = \sup_{p} \left[\langle p,\lambda\rangle - L(x,\lambda)\right].
\end{equation}
What $\langle\cdot\rangle$ means also depends on the context: on $\R^d$ it simply is
the inner product, while in general it is a natural pairing between a space and its
dual, such as $\langle f,\mu\rangle = \int f d\mu$.
\item
The Lagrangian in \eqref{pathspacelargedevpri} is the function $L$ with $x=\gamma_t$
and $\lambda =\dot{\gamma}_t$.
\end{enumerate}
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\end{document}
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