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\noindent{\Large \bf
%Title
A remark on the notion of robust phase transitions
%\break
\normalsize \rm}
\vspace{24pt}
\noindent{\bf Aernout C.D. van ENTER$^{1}$,
\vspace{12pt}
\it
\noindent
$^{(1)}$ Instituut voor theoretische natuurkunde R.U.G.
Nijenborgh 4, 9747AG, Groningen, the Netherlands.
\rm
\vspace{24pt}
\footnotesize
\begin{quote}
{\bf Abstract:}
We point out that the high-q Potts model on a regular lattice at its
transition temperature provides an example of a non-robust -- in the
sense recently proposed by Pemantle and Steif-- phase transition.
\vspace{3pt}
{\bf Keywords:} Robust phase transitions, Potts models
\end{quote}
\normalsize
\vspace{12pt}
%\section{Introduction}
%A.C.D. van Enter
In a recent paper \cite{PS} about phase transitions of spin models
on general trees, Pemantle and Steif introduced a
distinction between the notions of ordinary and "robust" phase transitions.
In their tree calculations, this notion of robust phase transitions
was argued somehow to be more natural. They proposed as an interesting
question to investigate this distinction also for regular lattices, and in
particular the possible occurrence of non-robust phase transitions for spin
models on regular lattices.
Here I point out that
at the transition temperature of a high-q Potts model on a regular lattice
the phase transition is non-robust, as a consequence of the occurrence of
a first-order transition in the temperature variable.
A phase transition (for Ising, Potts or n-vector models) is called robust
\cite{PS}, if weakening the bonds at the boundary of some large volume
to an arbitrarily small (but positive) strength in some non-symmetric state
does not influence the
non-symmetric character of the spin distribution at the origin, in the limit
where this boundary moves to infinity.
By this construction one can interpolate between pure and free boundary
conditions.
In the Ising model on a regular lattice robustness of the phase transition,
when it occurs, follows from a result of Lebowitz and Penrose\cite{lebpen},
that for all temperatures weak positive
boundary fields in the thermodynamic limit are equivalent to plus boundary
conditions, in the sense that both induce the same pure (plus) Gibbs measure.
This is in fact not so surprising, as below the critical temperature free (symmetric) boundary conditions
give rise to a symmetric mixture of pure ordered Gibbs measures, and
these mixtures are unstable for even a small asymmetry at the boundary.
After all, there are non-trivial --bounded-- observables at infinity by which
one can change the symmetric weight distribution of a mixture into an
asymmetric one, and the interpretation of such observables at infinity as
uniformly bounded, that
is, quite weak, boundary terms has been known for a considerable time
\cite{narrob}, see also \cite{brlepf} for related arguments).
Indeed, let $h$ be a non-trivial observable at infinity, such as exists
for (and only for) a mixed Gibbs measure, say $\mu$, then
\begin{equation}
\mu^{h}(.) = {{\mu(exp \ {h} \times .)}\over {\mu(exp \ {h})}} \neq \mu
\end{equation}
Thus the Gibbs measure (Gibbsian for the same interaction as $\mu$)
$\mu^{h}$ which is obtained formally by adding the bounded term
$h$ to the Hamiltonian, is different from $\mu$.
Moreover, in the references mentioned above it is described how $\mu^{h}$
can be
approximated by a sequence of measures of the form $\mu^{h_n}$ where the
$h_n$ form a uniformly bounded sequence of boundary terms associated to a
sequence of increasing volumes.
Thus it is not too
surprising that breaking the symmetry in the boundary conditions in the
manner described before, in which a small strength of a boundary field is
multiplied by the size of the boundary, has a more drastic effect because
this sequence of boundary terms
diverges, instead of staying uniformly bounded. Indeed, in the Ising model,
according to the Lebowitz-Penrose result it immediately drives
the state to be extremal.
We remark as an aside that the notion of robustness is intimately linked
with the single-spin space and the Hamiltonian having a symmetry
which gets broken by the weak boundary field.
In the high-q Potts case at the transition temperature, however, in contrast to
the low-temperature Ising case, there exists another
Gibbs measure having the Potts permutation symmetry,
beyond the symmetric mixture of the ordered measures. This is
the disordered state, which can be obtained by imposing free boundary
conditions, and which is pure, moreover\cite{KS,LMMRS,brkule}.
It takes
more "boundary (free) energy" to move from one pure state
to another than from a mixed state to one of its pure components.
Intuitively put, if one thinks of the pure states as valleys
in some (free)
energy landscape, to go from one pure state to another,
the system has to overcome a free energy barrier of boundary size.
This represents the cost of inserting a droplet of another (here an ordered)
phase. When the
boundary bonds are too weak, even if the state outside $\Gamma$ is ordered,
they can't provide sufficient energy to cross this
barrier between disordered and ordered phase, and favor order inside $\Gamma$.
On the other hand, mixed states are on top of this free energy
barrier, and they can be easily pushed off from there.
\smallskip
To be a bit more precise we now adapt the definition of Pemantle and Steif
to a regular lattice. (For the precise definitions, background and
general notions of Gibbs measure theory we refer to
\cite{Geo} or \cite{EFS_JSP}.)
Denote by $\mu_{J,\epsilon,\Gamma}^{+}$ the infinite-volume measure
which is obtained from the pure plus measure (we call the first of the q
Potts-states also ``plus'' here by abuse of terminology) by
multiplying the bond strengths $J$ in some contour $\Gamma$ by $\epsilon$.
$\Gamma$ is a set of bonds (or edges) such that their dual bonds (plaquettes)
form a closed (hyper-)surface separating the inside and outside of $\Gamma$.
It plays the role of the ``cutset'' of \cite{PS}. The choice of the plus
measure induces an effective boundary term, favoring order in
%-- not the same at all boundary sites, to be sure --
the plus direction, at the boundary $\Gamma$ of the volume $Int(\Gamma)$,
the strength of which is bounded above by $2d \times \epsilon \times \Gamma$.
\noindent
{\bf Remark:}
We remind the
reader that the (infinite-volume) plus measure is non-symmetric,
due to the assumption of
existence of a phase transition. Thus we have taken a first thermodynamic limit
already, and (by the DLR-equations) the marginal measure to the
configuration-space determined by the spins in the interior of $\Gamma$ is
an average over measures with different boundary conditions,
all with weak boundary bonds,
averaged with respect to this non-symmetric infinite-volume measure
$\mu_{J,\epsilon,\Gamma}^{+}$. As all boundary bonds in $\Gamma$ are weak,
this means that one can obtain the same measure by taking a
suitable weak boundary term added to the finite-volume Hamiltonian.
\noindent
{\bf Definition}:
A phase transition of a nearest neighbor Potts or n-vector model is robust if
for the marginal measure to the single-site space at the origin
for each positive $\epsilon \in (0,1]$, at least for some subsequence of
increasing contours $\Gamma_n$
whose interiors will finally include each finite volume,
\begin{equation}
%\lim_{\epsilon \to 0}
\lim_{\Gamma_n \to \infty} \mu_{J,\epsilon,\Gamma_n}^{+} \neq \mu_{J}^{free}.
\end{equation}
Our above discusion can be summarized in the following
\smallskip
\noindent
{\bf Theorem}:
For the the q-state Potts model on $Z^{d}$, d at least 2,
with q high enough, at the transition temperature the phase transition
is non-robust.
\smallskip
\noindent
{\bf Proof}:
We claim that for small enough $\epsilon$ the above inequality (2)
does not hold.
There are different ways one might approach a proof. We will sketch here
how one can adapt the Fortuin-Kasteleyn random-cluster representation
Pirogov-Sinai contour arguments of \cite {LMMRS} to
obtain one. In this random-cluster representation (for a detailed description
of the random-cluster representation for Potts models, see for
example \cite{FK,edwsok,ACCN,Gri,gehama}),
one considers an associated correlated edge-percolation
model, in which in a finite volume $\Lambda$ the probability of an
edge configuration $\eta \in (0,1)^{B(\Lambda)}$ is given by:
\begin{equation}
\mu^{\Lambda} (\eta) = {1 \over Z}{ \prod_{e \in B(\Lambda)}
p_e^{\eta_e} (1-p_e)^{1- \eta_e} q^{C(\eta)}},
\end{equation}
where $C(\eta)$ denotes the number of occupied connected clusters in the
configuration $\eta$, and
\begin{equation}
p_e = 1 - exp( \ -J_e) \ ,
\end{equation}
with $J_e$ the bond strength along edge $e$.
Percolation in the random-cluster model occurs if and only if
there is long-range order in the associated spin model.
First we notice that the finite-volume measures obtained by taking wired
boundary conditions outside a volume $\Lambda_m$ containing $\Gamma$
decrease, in FKG-sense, as the $\Lambda_n$
grow, and each of them FKG-dominates the marginal on $int(\Gamma)$ of the
infinite-volume wired measure on with ``weak'' bonds in $\Gamma$.
The wired boundary conditions correspond to having all edges occupied outside
the region, or, in spin language, to having all spins aligned (for instance in
the plus configuration). In particular this infinite-volume wired state
is associated to the measure $\mu^{+}_{J, \epsilon, \Gamma}$.
%Moreover,
%the marginals of these infinite-volume wired measures increase in FKG-sense
%with $\Gamma$.
These observations have several implications:
\begin{itemize}
\item[ (i) ] The limit in the left-hand side of (2)) always exists (in the
weak sense). We emphasize that it is, in fact, a double limit formed by first
taking for each $\Gamma_n$ wired boundary conditions outside an increasing
sequence of volumes $\Lambda_m$ containing (the for the moment fixed)
$\Gamma_n$, (this limit
exists as it is a limit of FKG-decreasing measures), and then
taking the limit $\Gamma_n \to \infty$.
\item [ (ii)] If instead we take a ``diagonal'' limit $\Gamma_n \to \infty$
with wired boundary conditions outside $\Gamma_n$, we will see that we
get a convergent
sequence. Its limit is a measure that dominates any subsequence
limit of the left-hand
side of (2) (as for each $\Gamma$ the element of the sequence dominates the
corresponding element of the sequence in (i) ).
It turns out that it approaches the FKG-minimal state, that
is, (for high q) the disordered state.
\item [ (iii) ] As the limit in (ii) is the FKG-minimal random-cluster state,
so is the limit in (i).
%Therefore, as the origin is not
%connected to infinity in this latter ``diagonal'' limit, and as
%this connection is
%an increasing event, its absence is implied for any measure which is smaller in
%FKG-sense.
\end{itemize}
%taking wired boundary conditions -- that is all edges outside the
%volume are occupied, corresponding to plus boundary conditions
%in spin language -- outside $\Gamma$
%stochastically --FKG-- dominates the marginal measure on the spins in $\Gamma$,
%of the infinite-volume wired (ordered) state (the finite-volume measure with
%free boundary conditions on the other hand is stochastically less).
%The wired boundary conditions
%result in fact in the maximal (in FKG-sense) measure. Thus it is sufficient
%to show the result that the origin is not connected to infinity, in the limit
%where $\Gamma$ goes to infinity, with wired boundary conditions. This then by
%standard arguments about the connection between the random-cluster model and
%the associated spin model implies that the spin distribution at the origin is
%symmetric for the limit of finite-volume measures with
%plus boundary conditions outside $\Gamma$. (If in the {\em maximal}
%limit-random-cluster measure
%no percolation occurs, it will not happen in any other random-cluster measures
%either).
%This means that instead of two infinite-volume limits we
%now need to consider only one limit, namely $\Gamma$ to infinity, with
%``weakly wired'' boundary conditions at $\Gamma$.
We can take the $\Gamma$'s to be the boundary of large squares in d=2,
or (hyper-)cubes in higher dimensions, with the origin at the center.
In the first version of this paper I sketched how this weakly wired
limit (ii) can be compared with (and can be shown to coincide with)
the limit of measures with free boundary conditions through
adaptation of existing proofs of the first-order Potts
transition. I emphasized the arguments in \cite{brkule} (see Appendix),
but also mentioned
the random-cluster version. After submitting this version,
%After the first submission of this paper, which contained a sketch of
%the adaptations one should put to some existing proofs of the first-order
%Potts transition, necessary to handle these weak boundary bonds,
%as compared
%with free boundary conditions,
R. Koteck\'y kindly informed me that a detailed
version of the necessary contour analysis for the random-cluster version,
essentially along this line, was worked out by I. Medved \cite{med}, and
included in his more general analysis of finite-size effects. In his
terminology, having wired boundary conditions outside $\Gamma$ and
weak enough
bonds in $\Gamma$, is an example of having ``disordering boundary conditions''.
In his Section (2.2) a range of $\epsilon$ which are disordering is determined.
With the above FKG-domination-argument, the statement of the
non-robustness becomes
indeed a corollary of his results. $\Box$
%Furthermore we notice that the fact that the bonds in $\Gamma$ are weak means
%that in the random-cluster model these edges are $0$ (empty) with
%large probability.
%\noindent
%This follows directly from a Pirogov-Sinai analysis. We noticed before that one has to control a Potts model in a sequence of volumes $\Gamma_n$ with a weak field at the boundary of $\Gamma_n$. One can either do the analysis along the lines of \cite{brkule}, once one notices that in the comparison between the restricted ensembles of ordered and disordered squares, once a square contains bonds which are part of $\Gamma$, its energy contribution to such a square cannot be more than ${1 \over {2(d-1)}} \times \epsilon J$ per unbroken bond in $\Gamma$. This means that one can choose the same restricted ensembles as in \cite{brkule}, in which ``good'' squares were either totally ordered
%(all 4 sites of the square having the same Potts spin) describing a
%square which is typical for one of the q ordered phases, or totally disordered
%(all 4 sites of the square having a different value of the Potts spin),
%describing a square which is typical for the disordered state.
%Now, however, one has also to consider
%ordered squares which have two bonds in $\Gamma$ to be ``bad'', while if
%the bonds of such squares are ordered on the outside of $\Gamma$ and
%the site(s) inside are disordered (having all their neighbors different-valued)
%they are good.
%Indeed, broken weak bonds contribute to the entropy, but hardly can gain
%on the energy, while unbroken weak bonds do not contribute
%more than $\epsilon$ to the energy, and nothing to the entropy.
%Whereas in \cite{brkule} the bad squares (neither ordered nor disordered)
%from which the free energy contours are built,
%are suppressed by a weight factor ${1 \over q}^{1 \over 4}$, for bad squares
%touching
%the boundary this suppression factor can be multiplied by at most
%$exp{4 \over {2(d-1)} \times \epsilon}$.
%However, the Peierls contour argument
%is robust as long as the Peierls condition holds, because each bad square
%still contributes a free energy term which is uniformly bounded below.
%Thus we can obtain a Peierls estimate of similar form as equation (3.34)
%of \cite{brkule}.
%Thus as for an elementary square deep
%inside $\Gamma$ to be ordered, it has to be surrounded by a (Pirogov-Sinai)
%contour of bad squares, we can conclude that in the limit where $\Gamma$
%approaches infinity we obtain the disordered state, which has the full
%Potts symmetry.
\smallskip
Although the proof thus rests on Medved's result, let me add some explanatory
remarks, see also the Appendix.
The proof goes essentially along the lines of \cite{LMMRS}, once one
notices that, because of the above, for small $\epsilon$ only a small fraction
of the boundary edges in $\Gamma$ (in the FK-representation \cite{ACCN,FK,
edwsok,Gri}) are occupied.
Thus the system acts essentially like the system with
free boundary conditions outside $\Gamma$
(up to a boundary term of order $\epsilon J$ $\times$ $|\Gamma|$ which is
small compared to the boundary free energy term necessary to induce
the wired state on the inside of $\Gamma$ which is of order $J \times \Gamma$).
Differently put, the ``weakly wired'' boundary conditions do
not influence the behavior in the infinite-volume --- that is now
infinite-$\Gamma$ --- limit, as compared to the free boundary conditions.
The reason is that the Peierls contour
estimate for the probability of finding an essentially
ordered region inside $\Gamma$ is exponentially small in $|\Gamma|$.
Multiplying by a term $exp(\epsilon J \Gamma)$ does not qualitatively
change this.
At this point it is essential that q is large enough,
and that one is at the transition temperature where the pure disordered state
coexists with the q ordered ones.
For the low temperature Ising model with
free boundary conditions outside $\Gamma$ the probability of finding
the system inside $\Gamma$ in an essentially plus configuration is $1 \over 2$,
and similarly for Potts models in the low temperature region this probability
is $1 \over q$. These probabilities follow from symmetry, and are not obtained
by contour estimates. It is here that the essential difference with the
non-robustness example occurs.
%It is known that for free boundary conditions
%the pure disordered state is obtained, in the limit where the
%interior of the contour $int(\Gamma)$ increases to include all sites.
%The proofs of this fact for free boundary conditions
%which one employs
%are mostly based on Pirogov-Sinai contour arguments in some form or other.
%For the random-cluster proof the following observations indicate
%why such contour arguments should still apply if one changes
%free boundary conditions to small-$\epsilon$ wired boundary conditions.
%First we remark that with large probability, the fraction of occupied edges
%in $\Gamma$ is not larger than $2 \epsilon$.
%Then we notice that a change
%in contour weights only applies to contours touching the boundary. For an edge
%touching the origin to be enclosed by such a contour, the contour has to have
%a size comparable to $\Gamma$, which means that the Peierls condition typically
%still holds, albeit with a slightly worse constant, as long as
%$\epsilon$ is small enough.
For an earlier analysis of a related situation of
changing ``pure-state'' to ``almost-pure-state'' boundary conditions in
a more symmetric set-up see \cite {ccf}.
%The ordered phase contains typically
%squares in which all four edges are occupied, the disordered phase typically
%has squares in which all four edges are empty. The connected components of
%the boundary of the sets of ordered and disordered squares
%the set of squares which are neither ordered nor disordered form the
%Pirogov-Sinai type contours.
\bigskip
\smallskip
\noindent
{\bf Comment 1:}
The mechanism which causes the transition to be non-robust, is the fact
that there is a first-order transition in temperature, such that at the
transition temperature there is coexistence between a higher-entropy,
lower-energy disordered
state of higher symmetry, and a number of lower-entropy higher-energy
states which are of lower symmetry. Thus in more complicated models,
one expects the transition also to be non-robust whenever
there is a first order transition in temperature, accompanied by the breaking
of a symmetry on the low temperature side of the phase transition.
\bigskip
\smallskip
\noindent
{\bf Comment 2:}
Although also for the high-q (in fact already for any value of q larger than
2) Potts model on trees the notions of ordinary and robust phase transitions do
not coincide, and although, moreover, the high-q Potts model on trees, just as
on regular lattices, typically shows a first-order transition in the
temperature \cite{PS}, the interpretation of the distinction between ordinary
and robust transitions seems somewhat different on trees. The separation
between boundary terms and volume terms is more questionable on trees, so an
interpretation in terms of a "boundary free energy" analysis does not seem
possible. Indeed, on trees there can be coexistence of two ordered (the plus
and minus) states with a disordered one, even for the Ising model, on a whole
intermediate temperature interval \cite{CCST,blruza,io}. For the Ising model on
trees, however, Pemantle and Steif have showed that the occurrence of a phase
transition and a robust phase transition always coincide.
\section{APPENDIX}
In this appendix I present a way to get an alternative derivation
of Medved's result.
In particular, I sketch how the arguments of the Pirogov-Sinai
proof by Bricmont, Kuroda and
Lebowitz \cite{brkule}, (BKL) should be adapted to handle the case of weak
boundary bonds.
We give, as in \cite{brkule}, the argument for d=2. We compare only the weight
of ``boundary squares'', that is squares inside $\Gamma$ such that
they touch at least one (and thus two or three) of the weak bonds, as all
other computations are unchanged.
The new element as compared to \cite{brkule} is that now an ordered
boundary square will be a contour square (in the sense of BKL).
Indeed, call a $E_{4}'$ those sites in $E_4$ (that means, they touch four
ordered bonds), which
touch at least one weak bond. Such a site has at least
${1 \over 4} \times {(1- \epsilon)}$ less
energy
%of two (or three ) ineffective u-bonds,
than it would have if none of the boundary bonds were weak, but still
contributes zero entropy.
This implies that even with ordered boundary conditions, an ordered
boundary square with four $E_4$-sites
has lower free energy compared to a disordered boundary square with four
$E_0$-sites.
The definition of contours in terms of irregular --or contour--
squares is then the same as in BKL.
Thus BKL eq (3.34), which estimates the partition function in volume $\Lambda$
for all
configurations compatible with a prescribed configuration of broken and
unbroken bonds, is replaced by
\begin{equation}
Z(\Lambda|\underline u, \underline b) \leq q^{(E_{0} + E_{4} -E_{4}')} q^{({3 \over 4} + \epsilon)( E_{1}+E_{2}+E_{3} +E_{4}')},
\end{equation}
with $\Lambda$ equal to $int(\Gamma)$,
and BKL eq (3.35), the inequality for the ratio of partition
functions with or without the constraint that contour $\gamma$ is present
in volume $\Lambda$ with boundary condition $\omega$,
which expresses that the Peierls condition applies,
\begin{equation}
Q(\gamma|\Lambda, \omega) \leq 2^{2 |\gamma|} q^{ - {{C \over 4}|\gamma|}}
\end{equation}
still holds, but with a slightly worse constant C.
(As I already used the symbol $\Gamma$, I slightly changed the notation
of \cite{brkule} to
let $\gamma$ denote a contour).
For a central square to be ordered, it needs to be surrounded by a contour,
thus, at sufficiently high q and small enough (dependent on q) $\epsilon$,
this is of low probability, uniformly in the size of the enclosing boundary.
\section*{Acknowledgments}
I learned about the notion of robust phase transitions from
the Kac seminar lectures of Jeff Steif, whom I also thank for providing me
with the preprint of \cite{PS}. Some useful conversations with Marek
Biskup are gratefully acknowledged. I thank M. Biskup, R. Fern\'andez and
M. Winnink for some advice on the manuscript. I thank R. Koteck\'y for
informing me of the existence of \cite{med}, and explaining to me how the
necessary estimates are contained in Medved's work and R. Koteck\'y and
M. Biskup for making a copy available to me,
%\section{Basic Set-up}
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