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\begin{center}
{\footnotesize Available at:
{\tt http://publications.ictp.it}}\hfill IC/2010/035\\
\vspace{1cm}
United Nations Educational, Scientific and Cultural Organization\\
and\\
International Atomic Energy Agency\\
\medskip
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS\\
\vspace{2.5cm}
{\bf ON $P$-ADIC QUASI GIBBS MEASURES\\
FOR $Q+1$-STATE POTTS MODEL ON THE CAYLEY TREE}\\
\vspace{2cm}
Farrukh Mukhamedov\footnote{Junior Associate of ICTP. far75m@yandex.ru}\\
{\it Department of Computational and Theoretical Sciences,\\
Faculty of Science, International Islamic University Malaysia,\\
P.O. Box 141, 25710 Kuantan, Pahang, Malaysia\\
and\\
The Abdus Salam International Centre for Theoretical Physics,
Trieste, Italy.}
\end{center}
\vspace{1cm}
\centerline{\bf Abstract}
\baselineskip=18pt
\bigskip
In the present paper we introduce a
new class of $p$-adic measures, associated with $q+1$-state Potts
model, called {\it $p$-adic quasi Gibbs measure}, which is totally
different from the $p$-adic Gibbs measure. We establish the
existence $p$-adic quasi Gibbs measures for the model on a Cayley
tree. If $q$ is divisible by $p$, then we prove the occurrence of a
strong phase transition. If $q$ and $p$ are relatively prime, then
there is a quasi phase transition. These results are totally
different from the results of [F.M.Mukhamedov, U.A. Rozikov,
\textit{Indag. Math. N.S.} {\bf 15}(2005) 85--100], since $q$ is
divisible by $p$, which means that $q+1$ is not divided by $p$, so
according to a main result of the mentioned paper, there is a unique
and bounded $p$-adic Gibbs measure (different from $p$-adic quasi
Gibbs measure).
\vfill
\begin{center}
MIRAMARE -- TRIESTE\\
June 2010\\
\end{center}
\vfill
%{\it
%Mathematics Subject Classification}: 46S10, 82B26, 12J12, 39A70, 47H10, 60K35.\\
%{\it Key words}: $p$-adic numbers, Potts model; $p$-adic quasi
%Gibbs measure, phase transition.
\newpage
\setcounter{footnote}{0}
\section{introduction}
Due to the assumption that $p$-adic numbers provide a more exact and
more adequate description of microworld phenomena, starting in the
1980s, various models described in the language of $p$-adic analysis
have been actively studied
\cite{ADFV},\cite{FO},\cite{MP},\cite{V1}. The well-known studies in
this area are primarily devoted to investigating quantum mechanics
models using equations of mathematical physics \cite{ADV,V2,VVZ}.
Furthermore, numerous applications of the $p$-adic analysis to
mathematical physics have been proposed in
\cite{ABK},\cite{Kh1},\cite{Kh2}. One of the first applications of
$p$-adic numbers in quantum physics appeared in the framework of
quantum logic in \cite{BC}. This model is especially interesting for
us because it could not be described by using conventional real
valued probability. Besides, it is also known
\cite{Kh2,Ko,MP,Ro,Vi,VVZ} that a number of $p$-adic models in
physics cannot be described using ordinary Kolmogorov's probability
theory. After that in \cite{KYR} an abstract $p$-adic probability
theory was developed by means of the theory of non-Archimedean
measures \cite{Ro}. Using that measure theory in \cite{KL},\cite{Lu}
the theory of stochastic processes with values in $p$-adic and more
general non-Archimedean fields having probability distributions with
non-Archimedean values, has been developed. In particular, a
non-Archimedean analogue of the Kolmogorov theorem was proven (see
also \cite{GMR}). Such a result allows us to construct wide classes
of stochastic processes using finite dimensional probability
distributions\footnote{We point out that stochastic processes in the
field $\bq_p$ of $p$-adic numbers with values of real numbers have
been studied by many authors, for example,
\cite{AK,AZ1,AZ2,DF,Koc,Y}. In those investigations wide classes of
Markov processes over $\bq_p$ were constructed and studied. In our
case the situation is different, since probability measures take
their values in $\bq_p$. This leads our investigation to some
difficulties. For example, there is no information about the
compactness of $p$-adic values probability measures. }. Therefore,
this result gives us a possibility to develop the theory of
statistical mechanics in the context of the $p$-adic theory, since
it lies on the basis of the theory of probability and stochastic
processes. Note that one of the central problems of such a theory is
the study of infinite-volume Gibbs measures corresponding to a given
Hamiltonian, and a description of the set of such measures. In most
cases such an analysis depends on specific properties of
Hamiltonian, and a complete description is often a difficult
problem. This problem, in particular, relates to a phase transition
of the model (see \cite{G}).
The aim of this paper is devoted to the development of $p$-adic
probability theory approaches to study $q+1$-state nearest-neighbor
$p$-adic Potts model on a Cayley tree (see \cite{W}). We are
especially interested in the construction of $p$-adic quasi Gibbs
measures for the mentioned model, since such measures present more
natural concrete examples of $p$-adic Markov processes (see
\cite{KL}, for definitions). In \cite{MR1,MR2} we have studied
$p$-adic Gibbs measures and the existence of phase transitions for
the $q$-state Potts models on the Cayley tree\footnote{The classical
(real value) counterparts of such models were considered in
\cite{W}}. It was established that a phase transition occurs
\footnote{Here the phase transition means the existence of two
distinct $p$-adic Gibbs measures for the given model.} if $q$ is
divisible by $p$. This shows that the transition depends on the
number of spins $q$.
In the present paper we introduce a new class of $p$-adic measures,
associated with $q+1$-state Potts model, called {\it $p$-adic quasi
Gibbs measure}, which is totaly different from the $p$-adic Gibbs
measures considered in \cite{MR1}. In Section 3, we establish the
existence of $p$-adic quasi Gibbs measures for the said model on a
Cayley tree of order two, and give concepts of {\it strong phase
transition}, {\it phase transition} and {\it quasi phase transition}
for the given model in terms of $p$-adic quasi Gibbs measures. Later
on, in Section 4, we shall prove the occurrence of the strong phase
transition, whenever $q$ is divisible by $p$. When $q$ and $p$ are
relatively prime, then we establish the existence of the quasi phase
transition. These results totally different from the results of
\cite{MR1,MR2}, since $q$ is divisible by $p$ means that $q+1$ is
not divided by $p$, which according to \cite{MR1} means that
uniqueness and boundedness of $p$-adic Gibbs measure.
\section{preliminaries}
In what follows $p$ will be a fixed prime number, and $\bq_p$
denotes the field of $p$-adic field, formed by completing $\bq$ with
respect to the unique absolute value satisfying $|p|_p = 1/p$. The
absolute value $|\cdot|_p$, is non-Archimedean, meaning that it
satisfies the ultrametric triangle inequality $|x + y|_p \leq
\max\{|x|_p, |y|_p\}$.
Any $p$-adic number $x\in\bq_p$, $x\neq 0$ can be uniquely represented in the form
\begin{equation}\label{canonic}
x=p^{\g(x)}(x_0+x_1p+x_2p^2+...),
\end{equation}
where $\g=\g(x)\in\bz$ and $x_j$ are integers, $0\leq x_j\leq p-1$,
$x_0>0$, $j=0,1,2,...$. In this case $|x|_p=p^{-\g(x)}$.
We recall that an integer $a\in \bz$ is called {\it a quadratic
residue modulo $p$} if the equation $x^2\equiv a(\textrm{mod $p$})$
has a solution $x\in \bz$.
\begin{lem}\label{quadrat} \cite{Ko} In order that the equation
$$
x^2=a, \ \ 0\neq a=p^{\g(a)}(a_0+a_1p+...), \ \ 0\leq a_j\leq p-1, \
a_0>0
$$
has a solution $x\in \bq_p$, it is necessary and sufficient that the
following conditions are fulfilled:
\begin{enumerate}
\item[(i)] $\g(a)$ is even;\\
\item[(ii)] $a_0$ is a quadratic residue modulo $p$ if $p\neq 2$, and
moreover,
$a_1=a_2=0$ if $p=2$.
\end{enumerate}
\end{lem}
Note the basics of $p$-adic analysis, $p$-adic mathematical physics
are explained in \cite{Ko,S,VVZ}.
Let $(X,\cb)$ be a measurable space, where $\cb$ is an algebra of
subsets $X$. A function $\m:\cb\to \bq_p$ is said to be a {\it
$p$-adic measure} if for any $A_1,\dots,A_n\subset\cb$ such that
$A_i\cap A_j=\emptyset$ ($i\neq j$) the equality holds
$$
\mu\bigg(\bigcup_{j=1}^{n} A_j\bigg)=\sum_{j=1}^{n}\mu(A_j).
$$
A $p$-adic measure is called a {\it probability measure} if
$\mu(X)=1$. A $p$-adic probability measure $\m$ is called {\it
bounded} if $\sup\{|\m(A)|_p : A\in \cb\}<\infty $. Note that, in
general, a $p$-adic probability measure need not be bounded
\cite{K3,KL,Ko}.
For more detailed information about $p$-adic measures we refer the
reader to \cite{K3},\cite{KhN},\cite{Ro}.
Recall that the Cayley tree $\Gamma^k$ of order $ k\geq 1 $ is an
infinite tree, i.e., a graph without cycles, and each vertex has
exactly $ k+1 $ edges. Let $\Gamma^k=(V, L)$, where $V$ is the set
of vertices of $ \Gamma^k$, $L$ is the set of edges of $ \Gamma^k$.
The vertices $x$ and $y$ are called {\it nearest neighbors} and they
are denoted by $l=$ if there exists an edge connecting them. A
collection of the pairs $,\dots,$ is called a {\it
path} from the point $x$ to the point $y$. The distance $d(x,y),
x,y\in V$, on the Cayley tree, is the length of the shortest path
from $x$ to $y$. Now fix $ x^0 \in V $, and set
$$ W_n=\{x\in V| d(x,x^0)=n\}, \ \ \
V_n=\bigcup_{m=1}^n W_m, \ \ L_n=\{l=\in L | x,y\in V_n\}.
$$ The set of {\it direct successors} of $x$ is defined by
\begin{equation}\label{S(x)}
S(x)=\{y\in W_{n+1} : d(x,y)=1 \}, \ \ x\in W_n.
\end{equation}
Observe that any vertex $x\neq x^0$ has $k$ direct successors and
$x^0$ has $k+1$.
\section{ $p$-adic Potts model and its $p$-adic quasi Gibbs measures}
In this section we consider the $p$-adic Potts model where spin takes values in the
set $\Phi=\{0,1,2,\cdots,q\}$, here $q\geq 1$, ($\Phi$ is called a
{\it state space}) and is assigned to the vertices of the tree
$\G^k=(V,\Lambda)$. A configuration $\s$ on $V$ is then defined as a
function $x\in V\to\s(x)\in\Phi$; in a similar manner one defines
configurations $\s_n$ and $\w$ on $V_n$ and $W_n$, respectively. The
set of all configurations on $V$ (resp. $V_n$, $W_n$) coincides with
$\Omega=\Phi^{V}$ (resp. $\Omega_{V_n}=\Phi^{V_n},\ \
\Omega_{W_n}=\Phi^{W_n}$). One can see that
$\Om_{V_n}=\Om_{V_{n-1}}\times\Om_{W_n}$. Using this, for given
configurations $\s_{n-1}\in\Om_{V_{n-1}}$ and $\w\in\Om_{W_{n}}$ we
define their concatenations by
$$
(\s_{n-1}\vee\w)(x)= \left\{
\begin{array}{ll}
\s_{n-1}(x), \ \ \textrm{if} \ \ x\in V_{n-1},\\
\w(x), \ \ \ \ \ \ \textrm{if} \ \ x\in W_n.\\
\end{array}
\right.
$$
It is clear that $\s_{n-1}\vee\w\in \Om_{V_n}$.
The Hamiltonian $H_n:\Om_{V_n}\to\bq_p$ of the $p$-adic {\it
$q+1$-state Potts} model has the form
\begin{equation}\label{Potts}
H_n(\s)=N\sum_{\in L_n}\delta_{\s(x),\s(y)}, \ \
\s\in\Om_{V_n}, \ n\in\mathbb{N},
\end{equation}
where $\delta$ is the Kronecker symbol and the coupling constant $N$
belongs to $\bz$.
Note that when $q=1$, then the corresponding model reduces to the
$p$-adic Ising model. Such a model was investigated in \cite{GMR,KM}.
Now let us construct $p$-adic quasi Gibbs measures corresponding to the
model.
Assume that $\h: V\setminus\{x^{(0)}\}\to\bq_p^{\Phi}$ is a function,
i.e. $\h_x=(h_{0,x},h_{1,x},\dots,h_{q,x})$, where $h_{i,x}\in\bq_p$ ($i\in\Phi$) and $x\in V\setminus\{x^{(0)}\}$. Given $n\in\bn$, let us consider a $p$-adic probability measure $\m^{(n)}_\h$ on $\Om_{V_n}$ defined by
\begin{equation}\label{mu}
\mu^{(n)}_{\h}(\s)=\frac{1}{Z_n^{(\h)}}p^{H_n(\s)}\prod_{x\in
W_n}h_{\s(x),x}
\end{equation}
Here, $\s\in\Om_{V_n}$, and $Z_n^{(\h)}$ is the corresponding
normalizing factor called a {\it partition function} given by
\begin{equation}\label{ZN1}
Z_n^{(\h)}=\sum_{\s\in\Omega_{V_n}}p^{H_n(\s)}\prod_{x\in
W_n}h_{\s(x),x},
\end{equation}
here subscript $n$ and superscript $(\h)$ are accorded to the $Z$,
since it depends on $n$ and a function $\h$.
One of the central results of the theory of probability concerns a
construction of an infinite volume distribution with given
finite-dimensional distributions, which is called {\it Kolmogorov's
Theorem} \cite{Sh}. Therefore, in this paper we are interested in
the same question but in a $p$-adic context. More exactly, we want
to define a $p$-adic probability measure $\m$ on $\Om$ which is
compatible with defined ones $\m_\h^{(n)}$, i.e.
\begin{equation}\label{CM}
\m(\s\in\Om: \s|_{V_n}=\s_n)=\m^{(n)}_\h(\s_n), \ \ \ \textrm{for
all} \ \ \s_n\in\Om_{V_n}, \ n\in\bn.
\end{equation}
In general, \`{a} priori the existence of such a kind of measure
$\m$ is not known, since there is not much information on
topological properties, such as compactness, of the set of all
$p$-adic measures defined even in compact spaces\footnote{In the
real case, when the state space is compact, then the existence
follows from the compactness of the set of all probability measures
(i.e. Prohorov's Theorem). When the state space is non-compact, then
there is a Dobrushin's Theorem \cite{Dob1,Dob2} which gives a
sufficient condition for the existence of the Gibbs measure for a
large class of Hamiltonians. }. Note that certain properties of the
set of $p$-adic measures have been studied in \cite{kas2}, but those
properties are not enough to prove the existence of the limiting
measure. Therefore, at present, we can only use the $p$-adic
Kolmogorov extension Theorem (see \cite{GMR},\cite{KL}) which is
based on the so-called {\it compatibility condition} for the
measures $\m_\h^{(n)}$, $n\geq 1$, i.e.
\begin{equation}\label{comp}
\sum_{\w\in\Om_{W_n}}\m^{(n)}_\h(\s_{n-1}\vee\w)=\m^{(n-1)}_\h(\s_{n-1}),
\end{equation}
for any $\s_{n-1}\in\Om_{V_{n-1}}$. This condition, according to the
theorem, implies the existence of a unique $p$-adic measure $\m$
defined on $\Om$ with a required condition \eqref{CM}. Note that a
more general theory of $p$-adic measures has been developed in
\cite{kas1}.
So, if for some function $\h$ the measures $\m_\h^{(n)}$ satisfy
the compatibility condition, then there is a unique $p$-adic
probability measure, which we denote by $\m_\h$, since it depends on
$\h$. Such a measure $\m_\h$ is said to be {\it a $p$-adic quasi
Gibbs measure} corresponding to the $p$-adic Potts model. By
$Q\cg(H)$ we denote the set of all $p$-adic quasi Gibbs measures
associated with functions $\h=\{\h_x,\ x\in V\}$. If there are at
least two distinct $p$-adic quasi Gibbs measures $\m,\n\in Q\cg(H)$
such that $\m$ is bounded and $\n$ is unbounded, then we say that
{\it a phase transition} occurs. In other words, one can find two
different functions $\sb$ and $\h$ defined on $\bn$ such that there
exist the corresponding measures $\m_\sb$ and $\m_\h$, for which one
is bounded, and the other is unbounded. Moreover, if there is a
sequence of sets $\{A_n\}$ such that $A_n\in\Om_{V_n}$ with
$|\m(A_n)|_p\to 0$ and $|\n(A_n)|_p\to\infty$ as $n\to\infty$, then
we say that there occurs a {\it strong phase transition}. If there
are two different functions $\sb$ and $\h$ defined on $\bn$ such
that there exist the corresponding measures $\m_\sb$, $\m_\h$, and
they are bounded, then we say there is a {\it quasi phase
transition}.
\begin{rem} Note that in \cite{MR1} we considered the following
sequence of $p$-adic measures defined by
\begin{equation}\label{mr-mu}
\mu^{(n)}_{\h}(\s)=\frac{1}{\tilde Z_n^{(\h)}}\exp_p\{H_n(\s)\}\prod_{x\in
W_n}h_{\s(x),x},
\end{equation}
here as usual $\tilde Z_n^{(\h)}$ is the corresponding normalizing
factor. A limiting $p$-adic measure generated by \eqref{mr-mu} was
called {\it $p$-adic Gibbs measure}. Such kinds of measures and
phase transitions, for Ising and Potts models on Cayley tree, have
been studied in \cite{GMR,KM,MR1,MR2}. When a state space $\Phi$ is
countable, the corresponding $p$-adic Gibbs measures have been
investigated in \cite{KMM,M}.
\end{rem}
Now one can ask for what kind of functions $\h$ the measures
$\m_\h^{(n)}$ defined by \eqref{mu} would satisfy the
compatibility condition \eqref{comp}. The following theorem gives
an answer to this question.
\begin{thm}\label{comp1} The measures $\m^{(n)}_\h$, $
n=1,2,\dots$ (see \eqref{mu}) satisfy the compatibility condition
\eqref{comp} if and only if for any $n\in \bn$ the following
equation holds:
\begin{equation}\label{eq1}
\hat h_{x}=\prod_{y\in S(x)}{\mathbf{F}}(\hat \h_{y};\theta),
\end{equation}
here and below $\theta=p^N$, a vector $\hat \h=(\hat h_1,\dots,\hat
h_q)\in\bq_p^q$ is defined by a vector
$\h=(h_0,h_1,\dots,h_q)\in\bq_p^{q+1}$ as follows
\begin{equation}\label{H}
\hat h_i=\frac{h_i}{h_0}, \ \ \ i=1,2,\dots,q
\end{equation}
and mapping ${\mathbf{F}}:\bq_p^{q}\times\bq_p\to\bq_p^q$ is defined
by ${\mathbf{F}}(\xb;\t)=(F_1(\xb;\t),\dots,F_q(\xb;\t))$ with
\begin{equation}\label{eq2}
F_i(\xb;\theta)=\frac{(\theta-1)x_i+\sum\limits_{j=1}^{q}x_j+1}
{\sum\limits_{j=1}^{q}x_j+\theta}, \ \ \xb=\{x_i\}\in\bq_p^q, \ \
i=1,2,\dots,q.
\end{equation}
\end{thm}
The proof consists of checking condition \eqref{comp} for the
measures \eqref{mu} (cp. \cite{MR1,KMM}).
\begin{lem}\label{parti} Let $\h$ be a solution of \eqref{eq1}, and
$\m_\h$ be an associated $p$-adic quasi Gibbs measure. Then for the
corresponding partition function $Z^{(\h)}_n$ (see \eqref{ZN1})
the following equality holds
\begin{equation}\label{ZN2}
Z^{(\h)}_{n+1}=A_{\h,n}Z^{(\h)}_n,
\end{equation}
where $A_{\h,n}$ will be defined below (see \eqref{aN3}).
\end{lem}
\begin{proof} Since $\h$ is a solution of \eqref{eq1}, then we conclude that there is a constant
$a_\h(x)\in\bq_p$ such that
\begin{equation}\label{aN1}
\prod_{y\in
S(x)}\sum_{j=0}^qp^{N\d_{ij}}h_{j,y}=a_{\h}(x)h_{i,x}
\end{equation}
for any $i\in\{0,\dots,q\}$. From this one gets
\begin{eqnarray}\label{aN2}
\prod_{x\in W_{n}}\prod_{y\in
S(x)}\sum_{j=0}^q p^{N\d_{ij}}h_{j,y}=\prod_{x\in
W_n}a_{\h}(x)h_{i,x}=A_{\h,n}\prod_{x\in W_n}h_{i,x},
\end{eqnarray}
where
\begin{equation}\label{aN3}
A_{\h,n}=\prod_{x\in W_n}a_{\h}(x).
\end{equation}
Given $j\in\Phi$, by $\eta^{(j)}\in\Om_{W_n}$ we denote a configuration on $W_n$ defined as follows: $\eta^{(j)}(x)=j$ for all $x\in W_n$.
Hence, by \eqref{mu},\eqref{aN2} we have
\begin{eqnarray*}
1&=&\sum_{\s\in\Om_n}\sum_{\w\in\Om_{W_n}}\m^{(n+1)}_\h(\s\vee \w)\\
&=&\sum_{\s\in\Om_n}\sum_{\w\in\Om_{W_n}}\frac{1}{Z^{(\h)}_{n+1}}p^{H(\s\vee \w)}\prod_{x\in
W_{n+1}}h_{\w(x),x}\\
&=&\frac{1}{Z^{(\h)}_{n+1}}\sum_{\s\in\Om_n}p^{H(\s)}\prod_{x\in W_n}\prod_{y\in S(x)}\sum_{j=0}^q p^{N\d_{\s(x),j}}h_{j,y}\\
&=&\frac{A_{\h,n}}{Z^{(\h)}_{n+1}}
\sum_{\s\in\Om_n}p^{H(\s)}\prod_{x\in W_n}h_{\s(x),x}\\
&=&\frac{A_{\h,n}}{Z^{(\h)}_{n+1}}Z_n^{(\h)}
\end{eqnarray*}
which implies the required relation.
\end{proof}
\section{Existence of $p$-adic quasi Gibbs measures}
In this section we will establish existence of $p$-adic quasi Gibbs
measures on a Cayley tree of order 2, i.e. $k=2$. To do it, due to
Theorem \ref{comp1} it is enough to show the existence of a solution
of \eqref{eq1}.
Recall that a function $\h=\{\h_x\}_{x\in V\setminus\{x^0\}}$ is
called {\it translation-invariant} if $\h_x=\h_y$ for all $x,y\in
V\setminus\{x^0\}$. A $p$-adic measure $\m_\h$, corresponding to a
translation-invariant function $\h$, is called a {\it
translation-invariant $p$-adic quasi Gibbs measure}.
In what follows, we restrict ourselves to the description of
translation-invariant solutions of \eqref{eq1}, namely
$\h_x=\h(=(h_0,h_1,\dots,h_q))$ for all $x\in V$. Then \eqref{eq1}
can be rewritten as follows
\begin{equation}\label{eq11}
\hat h_{i}=\bigg(\frac{(\theta-1)\hat h_{i}+\sum_{j=1}^{q}\hat
h_j+1} {\sum_{j=1}^{q}\hat h_j+\theta}\bigg)^2, \ \ i=1,2,\dots,q.
\end{equation}
One can see that $(\underbrace{1,\dots,1,h}_m,1,\dots,1)$ is an
invariant line for \eqref{eq11} ($m=1,\dots,q$). On such kind of
invariant line equation \eqref{eq11} reduces to the following one
\begin{equation}\label{eq12}
u=\bigg(\frac{\theta u+q} {u + \t+q-1}\bigg)^2.
\end{equation}
A simple calculation shows that the last equality has the form
$$
(u-1)(u^2+(2\t-\t^2+2q-1)u+q^2)=0.
$$
Hence, $u_0=1$ solution defines a $p$-adic quasi Gibbs measure
$\m_0$.
Now we are interested in finding other solutions of \eqref{eq12},
which means we need to solve the following one
\begin{equation}\label{eq13}
u^2+(2\t-\t^2+2q-1)u+q^2=0.
\end{equation}
Observe that the solution of \eqref{eq13} can be formally written by
\begin{equation}\label{eq14}
u_{1,2}=\frac{-(2\theta-\theta^2+2q-1)\pm(\theta-1)\sqrt{D(\t,q)}}{2},
\end{equation}
where $D(\t,q)=\theta^2-2\theta-4q+1$
So, if the defined solutions exist in $\bq_p$, then they define $p$-adic quasi Gibbs measures $\m_{1}$ and
$\m_2$, respectively. Note that to exist such solutions the expression $\sqrt{D(\t,q)}$ should have
a sense in $\bq_p$, since in $\bq_p$ not every quadratic equation has a
solution (see Lemma \ref{quadrat}). Therefore, we are going to check
when $\sqrt{D(\t,q)}$ does exist.
Throughout the paper we will assume that $N>0$, this means
$|\t|_p=p^{-N}<1$. Now let us consider several distinct cases with
respect to $q$.
{\sc Case $q=1$}. Note that this case corresponds to the $p$-adic
Ising model, and $D(\t,1)=\theta^2-2\theta-3$.
\begin{enumerate}
\item[(i)] Let $p=2$. Then from $-3=1+2^2+2^3+\cdots$ one has
$$
D(\t,1)=1+2^2+2^3+2^4\e-2\t+\t^2,
$$
where $\e=1+2+2^2+\cdots$. Hence, due to Lemma \ref{quadrat} one can
check that for any $N\geq 1$ the $\sqrt{D(\t,1)}$ does not exist.
\item[(ii)] Let $p=3$. Then taking into account that $\t=p^N$ we find
$$
D(\t,1)=3(3^{2N-1}-2\cdot 3^{N-1}-1).
$$
If $N=1$ then $D(\t,1)=0$, so $\sqrt{D(\t,1)}$ exists. If $N>1$ then
due to Lemma \ref{quadrat} we conclude that $\sqrt{D(\t,1)}$ does
not exist.
\item[(iii)] Let $p\geq 5$. Then from the expression
$$-3=p-3+(p-1)p+(p-1)p^2+\cdots
$$
we obtain
$$
D(\t,1)=p-3+(p-1)p\e_1-2p^N+p^{2N},
$$
where $\e_1=1+p+p^2+\cdots$. So, according to Lemma \ref{quadrat}
$\sqrt{D(\t,1)}$ exists if and only if the equation $x^2\equiv p-3
(\textrm{mod} p)$ has a solution in $\bz$. It is easy to see that
the last equation equivalent to $x^2+3\equiv 0(\textrm{mod} p)$. For
example, when $p=7$ the equation $x^2+3\equiv 0(\textrm{mod} p)$ has
a solution $x=2$. So, in this case $\sqrt{D(\t,1)}$ exists.
\end{enumerate}
Hence, we can formulate the following
\begin{thm}\label{q1} Let $N\geq 1$ and $q=1$ (Ising model). Then the following
assertions hold true:
\begin{enumerate}
\item[(i)] If $p=2$, then there is a unique translation-invariant $p$-adic quasi Gibbs measure
$\m_0$;\\
\item[(ii)] Let $p=3$. If $N=1$, then there are three translation-invariant $p$-adic quasi Gibbs measures $\m_0$, $\m_1$ and $\m_2$, otherwise there is a unique translation-invariant $p$-adic quasi Gibbs measure $\m_0$;
\item[(iii)] Let $p\geq 5$, then there are three translation-invariant $p$-adic quasi Gibbs measures $\m_0$, $\m_1$ and $\m_2$ if and only if $-3$ is a quadratic residue of modulo $p$, otherwise there is a unique translation-invariant $p$-adic quasi Gibbs measure $\m_0$;
\end{enumerate}
\end{thm}
{\sc Case $q\geq 2$}. This case corresponds to $q+1$-state Potts
model. Here we shall consider several cases with respect to $p$.
\begin{enumerate}
\item[(i)] Let $p=2$. Let us represent $q$ in a 2-adic form, i.e.
$$
q=k_0+k_12+\cdots+k_s2^s, \ \ \ s \geq 1.
$$
Then we have
$$
-4q=2^2\big((2-k_0)+(1-k_1)2+\cdots+(1-k_s)2^s\big).
$$
Therefore, one has
$$
D(\t,q)=1+2^2\big((2-k_0)+(1-k_1)2+\cdots+(1-k_s)2^s\big)-2^{N+1}+2^{2N}.
$$
Now according to Lemma \ref{quadrat} we conclude that
$\sqrt{D(\t,q)}$ exists if and only if $k_0=0$, which is equivalent
to $|q|_2\leq 1/2$.
\item[(ii)] Let $p=3$. We represent $q$ in a 3-adic form, i.e.
$$
q=k_0+k_13+\cdots+k_s3^s, \ \ \ s \geq 0.
$$
Then we have
\begin{eqnarray*}
D(\t,q)&=&1-q-q\cdot 3-2\cdot 3^N+3^{2N}\nonumber \\
&=&1+(3-k_0)+(2-k_1)3+\cdots(2-k_s)3^s-q\cdot 3-2\cdot 3^N+3^{2N}.
\end{eqnarray*}
If $k_0=0$, then from Lemma \ref{quadrat} one can see that
$\sqrt{D(\t,1)}$ exists.
If $k_0=2$, then $\sqrt{D(\t,q)}$ does not exist, since $x^2\equiv
2(\mod 3)$ has no solution in $\bz$.
If $k_0=1$, then this case is more complicated. We cannot provide
any certain rule to check the existence of $\sqrt{D(\t,q)}$. But in
this case, $\sqrt{D(\t,q)}$ may exist or may not. For example, if
$k_1\neq 2$ then $\sqrt{D(\t,q)}$ does not exist whenever $N\geq 3$.
If $k_1=2$ and $k_2=2$ then $\sqrt{D(\t,q)}$ exists whenever $N\geq
4$.
\item[(iii)] Let $p\geq 5$. Let us represent $q$ in a $p$-adic expression
$$q=k_0+k_1p+\cdots+k_sp^s, \ \ \ s \geq 0.
$$
Then we have
$$
D(\t,1)=1+4(p-k_0)+4(p-1-k_1)p+\cdots+4(p-1-k_s)p^s-2p^N+p^{2N}.
$$
So, according to Lemma \ref{quadrat} $\sqrt{D(\t,q)}$ exists if the
equation $x^2\equiv 1-4k_0(\mod p)$ has a solution in $\bz$ whenever
$1-4k_0$ is not divided by $p$. It is clear that if $k_0=0$ then the
equation has a solution for any value of $p$ ($p\geq 5$). Note that if $1-4k_0$ is divided by $p$, then $\sqrt{D(\t,q)}$ does not exist.
If $p=5$ and $k_0=3$, then one can check that $x^2\equiv
-11(\textrm{mod} 5)$ has a solution $x=5n+2$. So, in this case
$\sqrt{D(\t,q)}$ exists.
\end{enumerate}
So, we have the following
\begin{thm}\label{q2} Let $N\geq 1$ and $q\geq 2$ (Potts model). Then the following
assertions hold true:
\begin{enumerate}
\item[(i)] If $|q|_p<1$, then there are three translation-invariant $p$-adic quasi Gibbs measures $\m_0$, $\m_1$ and $\m_2$;
\item[(ii)] Let $p=3$. If $|q-2|_p<1$, then
there is a unique translation-invariant $p$-adic quasi Gibbs measure
$\m_0$; if $|q-1|_p<1$ there is at least one translation-invariant
$p$-adic quasi Gibbs measure $\m_0$;
\item[(iii)] Let $p\geq 5$ and $|4q-1|_p<1$, then there is a unique translation-invariant $p$-adic quasi Gibbs measure $\m_0$;
\end{enumerate}
\end{thm}
\section{Boundedness of $p$-adic quasi Gibbs measures and phase transitions}
In this section we study boundedness and unboundedness of the
$p$-adic quasi Gibbs measures $\m_0$,$\m_1$ and $\m_2$. In what
follows we consider a case $N\geq 1$, this means that $|\t|_p\leq
1/p$.
Assume that $\hat \h_1$, $\hat \h_2$ are solutions of \eqref{eq1}.
In what follows, without loss of generality, we may assume that
$\hat\h_1=(\hat h_1,1,\dots,1)$ and $\hat \h_2=(\hat
h_2,1,\dots,1)$, here $\hat h_i\neq 1$ and $\hat h_i$ ($i=1,2$) are
solutions of \eqref{eq13}. Therefore, we have
\begin{equation}\label{vieta-h}
\hat h_1+\hat h_2=-2q+1+\t^2-2\t, \ \ \ \hat h_1\cdot\hat h_2=q^2.
\end{equation}
Furthermore, we are going to consider the $p$-adic quasi Gibbs measures corresponding to these solutions.
Due to Lemma \ref{parti} the partition function $Z_{i,n}$ corresponding to the measure $\m_i$ ($i=1,2$) has the following form
\begin{equation}\label{Z-IN}
Z_{i,n}=a_i^{|V_{n-1}|}
\end{equation}
where $a_i=(\hat h_i+\t+q-1)^2h_0$.
For a given configuration $\s\in\Om_{V_n}$ denote
$$
\#\s=\{x\in W_n:\ \s(x)=1\}.
$$
From \eqref{mu},\eqref{H} and \eqref{Z-IN} we find
\begin{eqnarray}\label{e-mu1}
|\m_1(\s)|_p&=&\frac{1}{Z_{1,n}}\cdot\frac{1}{p^{H(\s)}}\prod_{x\in W_n}\bigg|\frac{h_{\s(x),x}}{h_0}\bigg|_p|h_0|_p^{|W_n|}\nonumber\\
&=&\frac{|h_0|_p^{|W_n|-|V_{n-1}|}}{|\hat h_1+\t+q-1|_p^{2|V_{n-1}|}}\cdot\frac{|\hat h_1|_p^{\#\s}}
{p^{H(\s)}}\nonumber\\
&=&\frac{|h_0|_p^2}{|\hat h_1+\t+q-1|_p^{2|V_{n-1}|}}\cdot\frac{|\hat h_1|_p^{\#\s}}{p^{H(\s)}},
\end{eqnarray}
where we have used the equality $|W_n|-|V_{n-1}|=2$.
Similarly, one gets
\begin{eqnarray}\label{e-mu2}
|\m_2(\s)|_p=\frac{|h_0|_p^2}{|\hat h_2+\t+q-1|_p^{2|V_{n-1}|}}\cdot\frac{|\hat h_2|_p^{\#\s}}{p^{H(\s)}},
\end{eqnarray}
Now assume that $q$ is divided by $p$, i.e. $|q|_p\leq 1/p$. Note
that according to Theorem \ref{q2} in the current case (i.e.
$|q|_p<1$) there exist the solutions $\hat h_1$ and $\hat h_2$.
Hence, from \eqref{vieta-h} we conclude that $|\hat h_1+\hat
h_2|_p=1$ and $|\hat h_1\cdot\hat h_2|_p=|q^2|_p\leq p^{-2}$. From
the last equality, without loss of generality, it yields that
\begin{equation}\label{n-h-12}
|\hat h_1|_p=|q^2|_p<1, \quad |\hat h_2|_p=1.
\end{equation}
Hence, we obtain $|\hat h_1+\t+q-1|_p=1$, therefore, from \eqref{e-mu1} one gets
\begin{equation}\label{e-mu1-2}
|\m_1(\s)|_p=\frac{|h_0|_p^2}{p^{H(\s)}}\cdot |\hat h_1|_p^{\#\s}\leq |h_0|_p^2,
\end{equation}
which implies that the measure $\m_1$ is bounded.
The equality \eqref{vieta-h} implies that
$\hat h_2-1=\t^2-2\t-2q-\hat h_1$, this with the strong triangle inequality and \eqref{n-h-12} yields
\begin{eqnarray}\label{h2-1}
|\hat h_2+\t+q-1|_p=|\t^2-\t-q-\hat h_1|_p=|q|_p,
\end{eqnarray}
if $|\t|_p\leq |q|^2_p$.
Hence, from \eqref{e-mu2} with \eqref{h2-1},\eqref{n-h-12} we find
\begin{eqnarray}\label{e-mu2-2}
|\m_2(\s)|_p&=&\frac{|h_0|_p^2}{|q|_p^{2|V_{n-1}|}}\cdot\frac{1}{p^{H(\s)}}\nonumber\\
&\geq& |h_0|_p^2p^{2|V_{n-1}|-H(\s)}
\end{eqnarray}
Now let us choose $\s_{0,n}\in\Om_{V_n}$ as follows $\s_{0,n}(x)=1$ for all $x\in V_n$. Then one can see that
$H(\s_{0,n})=0$, therefore from \eqref{e-mu2-2} one gets
$$
|\m_2(\s_{0,n})|_p\geq |h_0|_p^2p^{2|V_{n-1}|}\to\infty \ \ \textrm{as} \ \ n\to\infty.
$$
This yields that the measure $\m_2$ is not bounded.
Let us consider the measure $\m_0$. Similarly, we obtain
\begin{eqnarray}\label{e-mu0}
|\m_0(\s)|_p&=&\frac{|h_0|_p^2}{|\t+q|_p^{2|V_{n-1}|}}\cdot\frac{1}{p^{H(\s)}}\nonumber\\
&=&\frac{|h_0|_p^2}{|q|_p^{2|V_{n-1}|}}\cdot\frac{1}{p^{H(\s)}}\nonumber\\
&\geq& |h_0|_p^2p^{2|V_{n-1}|-H(\s)}
\end{eqnarray}
so, we immediately find that $|\m_0(\s_{0,n})|_p\to\infty$ as $n\to\infty$.
From \eqref{e-mu2-2}, \eqref{e-mu0} we immediately find
$$
\bigg|\frac{\m_0(\s)}{\m_2(\s)}\bigg|_p=1.
$$
Now let us compare $\m_1$ and $\m_2$. From \eqref{e-mu1-2},\eqref{e-mu2-2} with \eqref{n-h-12}
one finds
\begin{eqnarray}\label{e-mu12}
|\m_1(\s_{0,n})\m_2(\s_{0,n})|_p&=&\frac{|h_0|_p^4|\hat h_1|_p^{\#\s_{0,n}}}{|q|_p^{2|V_{n-1}|}}\nonumber\\
&=&|h_0|_p^4|q|_p^{2(|W_n|-|V_{n-1}|)}\nonumber\\
&=&|h_0|_p^4|q|_p^4.
\end{eqnarray}
This implies that $|\m_1(\s_{0,n})|_p\to 0$ as $n\to\infty$.
Consequently, we can formulate the following
\begin{thm}\label{bound1}
Let $N\geq 1$, $|q|_p<1$ and $|\t|_p\leq |q|^2_p$. Assume that $\m_0$,$\m_1$ $\m_2$ are $p$-adic
quasi Gibbs measures for the Potts model \eqref{Potts}. Then the measure $\m_1$ is bounded, while
the measures $\m_0$ and $\m_2$ are unbounded. Moreover, there is a strong phase transition.
\end{thm}
Now assume that $|q|_p=1$ and there exist solutions $\hat h_1$ and
$\hat h_2$. Note that, in general, the solutions may not exist (see
Theorems \ref{q1} and \ref{q2}). Then from \eqref{vieta-h} we find
that
\begin{eqnarray}\label{n-h-12-2}
|\hat h_1+\hat h_2|_p\leq 1,\\
\label{n-h-12-3}
|\hat h_1\cdot\hat h_2|_p=1.
\end{eqnarray}
In this case, one has $|\hat h_1|_p=1$ and $|\hat h_2|_p=1$. Indeed,
assume that $|\hat h_1|_p<1$, then the equality \eqref{n-h-12-3}
yields $|\hat h_2|_p>1$. Due to the strong triangle inequality we
get $|\hat h_1+\hat h_2|_p>1$ which contradicts \eqref{n-h-12-2}.
So, we have $|\t\hat h_i+q|_p=1$, since $|\t|_p<1$. On the other hand, we know that $\hat h_i$ ($i=1,2$) are solutions \eqref{eq12}, therefore, from \eqref{eq12} one gets
\begin{equation}\label{e-h-12}
|\hat h_i+\t+q-1|_p^2=\frac{|\t\hat h_i+q|_p^2}{|\hat h_i|_p}=1.
\end{equation}
Hence, \eqref{e-mu1}, \eqref{e-mu2} with \eqref{e-h-12} imply
\begin{equation}\label{e-m12}
|\m_i(\s)|_p=\frac{|h_0|_p^2|\hat h_i|_p^{\#\s^{(i)}}}{p^{H(\s)}}=
\frac{|h_0|_p^2}{p^{H(\s)}}\leq |h_0|^2_p \ \ \ (i=1,2).
\end{equation}
Let us consider the following difference
\begin{eqnarray}\label{e-dif-12}
|\m_0(\s)-\m_i(\s)|_p=\frac{|h_0|_p^2}{p^{H(\s)}}\bigg|(\t+q-1+\hat h_i)^{2|V_{n-1}|}-\hat h_i^{\#\s}(\t+q)^{2|V_{n-1}|}\bigg|_p.
\end{eqnarray}
Denoting
$$
x=\t+q-1, \ \ y=\hat h_i, \ \ N=2|V_{n-1}|, \ \ k=\#\s
$$
and taking into account $|x|_p\leq 1$ and $|y|_p=1$, the right-hand
side of \eqref{e-dif-12} can be estimated as follows
\begin{eqnarray}\label{e-dif-1}
|(x+y)^N-y^k(x+1)^N|_p&=&\bigg|\sum_{\ell=0}^NC_N^\ell x^\ell(y^{N-\ell}-y^k)\bigg|_p\nonumber\\
&=&\bigg|\sum_{\ell=0}^NC_N^\ell x^\ell y^{\min\{N-\ell,k\}}(1-y^{M_\ell})\bigg|_p\nonumber\\
&=&\bigg|(1-y)\sum_{\ell=0}^NC_N^\ell x^\ell y^{\min\{N-\ell,k\}}\big(\sum_{j=0}^{M_\ell}y^j\big)\bigg|_p\nonumber\\
&\leq& |1-y|_p\max\limits_{0\leq\ell\leq N}\bigg\{\bigg|C_N^\ell x^\ell y^{\min\{N-\ell,k\}}\big(\sum_{j=0}^{M_\ell}y^j\big)\bigg|_p\bigg\}\nonumber\\
&\leq&|1-y|_p,
\end{eqnarray}
here $M_\ell=\max\{N-\ell,k\}-\min\{N-\ell,k\}$.
From \eqref{e-dif-1} with \eqref{e-dif-12} we immediately find
\begin{eqnarray}\label{e-dif-2}
|\m_0(\s)-\m_i(\s)|_p\leq\frac{|h_0|_p^2|1-\hat h_i|_p}{p^{H(\s)}} \ \ \ (i=1,2).
\end{eqnarray}
Using the same argument one can find
\begin{eqnarray}\label{e-dif-3}
|\m_1(\s)-\m_2(\s)|_p\leq\frac{|h_0|_p^2|\hat h_1-\hat h_2|_p}{p^{H(\s)}}.
\end{eqnarray}
Consequently, we can formulate the following
\begin{thm}\label{bound1}
Let $N\geq 1$, $|q|_p=1$ and $\m_0$,$\m_1$ $\m_2$ be $p$-adic
quasi Gibbs measures for the Potts model \eqref{Potts}. Then the measures $\m_k$ ($k=0,1,2$) are bounded.
Moreover, the inequalities \eqref{e-dif-2},\eqref{e-dif-3} hold. In this case, there is a quasi phase transition.
\end{thm}
\section*{Acknowledgments} The present study has been done within
the grant FRGS0409-109 of Malaysian Ministry of Higher Education.
The final part of the present work was done while the author was
visiting the Abdus Salam International Centre for Theoretical
Physics, Trieste, Italy as a Junior Associate. He would like to
thank the Centre for hospitality and financial support.
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