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\title{The Ising model in a random field}
%\titlerunning{...}

\author{
{\normalsize Christof K\"ulske  }\\
{\normalsize Department of Mathematics and Computing Sciences} \\
{\normalsize University of Groningen},\\
{\normalsize Nijenborgh 9, }\\
{\normalsize 9747 AG,Groningen},\\
{\normalsize  the Netherlands}\\
\\[3 mm]
%\bigskip \\
}


\begin{document}
\maketitle

\newpage


\section{The Ising model in a random field}

The Ising model in a random field is a prominent example of a lattice spin model 
with quenched disorder. 
The question whether and when quenched disorder destroys the Ising phase transition at low temperatures  
was under dispute among theoretical physicists for quite some time \cite{ImMa75,PaSo79}. 
It was finally resolved by 
two papers in mathematical physics proving ferromagnetic ordering 
in $3$ or more dimensions  at low temperature \cite{BrKu88}, 
 and absence of ferromagnetic ordering in $2$ dimensions at any temperature \cite{AiWe90}.  
 
The joint measures of the model on the product of spin-space and the space 
of the disorder variables violate the Gibbsian variational principle 
and provide an example of the breakdown of classical Gibbsian theory when one 
has only a weakly Gibbs representation \cite{KuLeRe04}. 

Moreover the random field Ising model on the lattice 
and its exactly solvable mean-field 
counterpart play a role as an important case study of a disordered system 
in the study of chaotic volume dependence (metastates),  
metastability, and stochastic time-evolutions of Gibbsian measures and their Gibbsian properties. 


%\subsubsection*{The model}

{\em The model.} The model is defined in terms of the 
formal Hamiltonian that associates to an infinite-volume spin configuration 
 $\s=(\s_i)_{i\in \Z^d}$ with $\s_i\in \{-1,1\}$ the formal energy 
 \begin{equation*}
H^\eta(\s)=-\sum_{\langle i,j \rangle}\s_i \s_j -h\sum_{i}\eta_i \s_i
\end{equation*}
Here the first sum is over pairs of nearest neighbors on the lattice. 
The minus sign provides for a ferromagnetic coupling making the spins 
prefer to be aligned in the same direction to lower the energy, like in the usual Ising model where the second term is absent. The parameter $h$ denotes the strength 
of the coupling to the random fields $\eta=(\eta_i)_{i\in \Z^d}$. 
A configuration of such random fields is drawn 
according to a symmetric, sitewise independent probability distribution $\P$  
and is kept fixed ("quenched") in the course of the analysis. Main examples 
of this are the standard Gaussian distribution and the symmetric Bernoulli 
distribution where the values plus and minus one are taken with the same probability 
one half. 

 As usual in quenched systems one is interested in properties that 
hold for {\em typical} realizations $\eta$, that is for properties that hold for a set of realizations 
of $\P$-measure one. 

Following the standard definition in mathematical equilibrium statistical mechanics, 
the {\it quenched Gibbs measures} $\mu[\eta]$ are those measures 
on infinite-volume spin configurations $\s$ that, for a fixed random field configuration 
$\eta$, at inverse temperature $\beta\geq 0$ are solutions to 
the Dobrushin-Lanford-Ruelle (DLR-)equations. 
 
This means that they are  those measures $\mu[\eta]$ 
whose conditional measures (obtained by conditioning to a configuration $\bar \s$ 
outside of a finite volume $\L\subset \Z^d$)  are given by the Boltzmann 
weights computed from the restriction of the Hamiltonian to $\L$, 
including the couplings to the boundary condition $\bar \s$.  

The fundamental question is to understand the nature of these measures, 
for a typical realization $\eta$, and whether there is more than one of them, in which 
case there is a phase transition. 

%More precisely, the probability to see $\s$ inside $\L$ conditional 
%that we see $\bar \s$ outside $\L$ is given by  
% \begin{equation*}
% \begin{split}
% \mu[\eta](\s_{\L} |\bar \s_{\L^c} )= \frac{1}{Z_{\L}[\eta]}
% \exp\bigl ( \beta  \sum_{{\langle i,j \rangle}\atop{ i,j \in \L}}\s_i \s_j  + \beta  \sum_{{\langle i,j \rangle}
% \atop{i\in \L, j\in \L^c}}\s_i\bar \s_j     + \beta h\sum_{i\in \L}\eta_i \s_i\bigr) 
%\end{split}
%\end{equation*}
%where $Z_{\L}[\eta]$ is the disorder-dependent normalization factor (the partition function) 
%necessary to create a probability measure on the right hand side. 

{\em Ordering at low temperatures, small disorder, in $d\geq 3$.}
\cite{BrKu88} showed that,  in the three- or more-dimensional random field Ising model, 
for a symmetric 
random field distribution, small coupling $h$, and large inverse temperature, 
disorder does not destroy the ferromagnetic ordering of the regular Ising model. 
There exist distinguished Gibbs measures
$\mu^+[\eta]$ (and $\mu^-[\eta]$) obtained as weak 
limits with $+$ (and $-$)-spin boundary conditions. 
The authors of \cite{[BrKu88]} showed that the following picture holds:   
$\mu^+[\eta]$, for a fixed typical magnetic field configuration $\eta$, 
has typical spin configurations $\s$ that look like 
small perturbations around a plus-like (respectively a minus-like) 
infinite-volume ground state. 
At fixed $\eta$, a  plus-like ground state itself looks like a sea of pluses with rare islands 
of minuses in those 
regions of space where the realizations of the magnetic fields happen to be mostly 
oriented to favor the minus spins. 


{\em The contour renormalization group.}
Here is the idea of the proof of \cite{BrKu88}, for a pedagogic exposition see also \cite{Bo06,BoKu94}. 
Given a spin 
configuration, contours are the boundaries of regions on the lattice where the spins 
have the same sign. 
A spin configuration of the model can be translated 
into a contour description.  A set of such contours has a probability weight that is directly 
inherited from the random field Hamiltonian. 
The situation would be simple and a phase transition could be proved at low enough temperatures 
if we had a model where there is an energetic cost for the formation of an additional single contour 
that is of the order of the length of the contour (Peierls estimate). 
The situation in the random model is difficult precisely 
because such an estimate is not uniformly true; 
in fact for some configurations of random fields the introduction of a contour 
will lower the energy in comparison to the (say) all plus state. 
 This is to say that already the ground state will contain contours. 
The situation is hopeful nevertheless, because in $d\geq 3$ a failure of a Peierls estimate in a given 
contour is unlikely (Imry-Ma argument). It is still very non-trivial because there are too many contours  to add up corresponding probabilities (entropy of contours).  

The method used to overcome this difficulty 
is a renormalization group in contour space. It is a suitably defined map between an initial space 
of contours to an image space of coarser contours (containing less information 
about the initial spin configuration).  
A spatial rescaling takes place to be able to interpret the map as acting between the same 
initial and final contour space. 
This map now will be iterated 
and gives rise to a flow on probability distributions on these contour spaces, 
with discrete "time"-variable given by the level of the iteration.
The control of the flow of these probability distributions is technically difficult.  
It turns out however that the relevant features of these 
probability distributions can be described by 
an effective (or renormalized) temperature and an effective (or renormalized) 
field. Both of these variables converge to zero (in distribution) under the iteration 
of the renormalization map in spatial dimensions $d\geq 3$. 
This shows that they become irrelevant on large length scales. 
Hence the model shows ferromagnetic ordering. 

Non-rigorous manipulations of perturbative expansions (the dimensional reduction
of Parisi and Sourlas) in the theoretical physics literature 
had previously led to a wrong prediction of the low-temperature behavior.  
Opposed to that, the contour renormalization group method is a rigorous solution to the  
entropy of contours-problem based on the ideology to construct self-maps.  
This ideology, however, is    
well-inspired by non-rigorous methods of field theory as applied to the study of criticial phenomena. 
 Extensions of the contour-renormalization group method for the random field Ising model 
have been used in the study of random interfaces in a random environment \cite{BoKu94}, 
the long-range random field Ising model  with Kac-potential,  and for 
 the $\R$-valued  spin model ($\phi^4$-theory) in the presence of external random fields \cite{Ku99}.  
 
 {\em Non-existence of ferromagnetic order in $d=2$.}
 It was proved in \cite{AiWe90} that 
there is uniqueness of the Gibbs measure in 2 dimensions, at any fixed temperature, for P-a.e. 
$\eta$.  It shows that randomness can potentially alter the behavior of the system in a fundamental 
way, and cannot always be treated as a small perturbation. The method of \cite{AiWe90} 
is based on getting lower estimates on the fluctuations w.r.t. the distribution $\P$ 
of differences of free energies in finite volume taken with different boundary 
conditions. This method uses a martingale convergence result from probability theory to show the 
persistence of fluctuations w.r.t. the underlying randomness of free energies obtained 
with different boundary conditions, 
under the assumption that the state $\mu^+(\eta)$ is different from $\mu^-(\eta)$. 
This leads to a contradiction to the a priori bounds on such fluctuations 
which one obtains from varying the boundary condition.  So it implies 
the impossibility of having different phases. 

%Another virtue of the paper is that it contained implicitly 
%the definition of the so-called Aizenman-Wehr metastate. 


%We remark that it was later applied by [BoK96] to show the non- 
%localization of interfaces in random environments in the framework of certain models for 
%interfaces without overhangs in space dimensions less than $3 = 2 + 1$.


{\em Joint measures and failure of variational principle.}
The joint measures of the random field Ising model (whose study is advocated by 
the so-called Morita-approach in theoretical physics) 
provide an example of the subtleties  that may occur in infinite-volume 
lattice spin measures.  
While the joint measures are built of Gibbsian measures they are not 
Gibbsian themselves, and very strongly not so. 
To define them, we define first the joint variable $\xi_i$ at lattice site $i$ 
to be the pair of spin variable $\s_i$ and random field variable $\eta_i$. 
In the case of the plus-minus random field this leaves us with $4$ possible values. 
Call the measure on the infinite-volume space of joint variables 
$K^+(d\xi)=\P(d\eta)\mu^+(\eta)(d\s)$. We put ourselves again at low temperature 
in $3$ dimensions. Then the following properties hold. 

1) The conditional probability under $K^+$ to see a configuration $\x_0$ at the origin 
given a joint configuration outside of the origin 
is different from that for $K^-$. 

2) The relative entropy density (or information gain per site) between $K^+$ and the measure $K^-$ vanishes (where $K^-$ is correspondingly defined with minus boundary conditions.) 

These two properties are a violation to 
the classical variational principle which states that the relative entropy density 
between two proper Gibbs measures (with uniformly absolutely 
summable Hamiltonian) only vanishes if they have the same conditional probabilities. 
More can be said about the conditional probabilities of the joint measures. 

3) They have a representation in terms 
of an exponential of a Hamiltonian as an infinite sum of local 
terms indexed by subsets of $\Z^d$ and this sum converges exponentially 
fast when one restricts to a set of full joint measure (weakly Gibbsian representation). 
It is impossible to extend it such that it converges absolutely 
uniformly for {\em all} joint configurations. 

It is important to mention this property since it shows that 
a weakly Gibbsian representation is in general not good enough to insure the 
validity of Gibbsian theory, as was conjectured before.  In fact, the joint measures 
are non-Gibbs in a strong sense (almost surely non-Gibbs) which means that:    

4) The set of continuity points (in the product topology) has joint measure zero.  
This happens although the set of points where the Hamiltonian 
converges has joint measure one.  


\begin{thebibliography}{}



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\end{thebibliography}


\end{document}
