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\begin{document}
\title{Partition function zeros of adsorbing self-avoiding walks}

\author{EJ Janse van Rensburg}

\address{Department of Mathematics \& Statistics, York University, Toronto, Ontario, M3J 1P3, Canada}

\ead{rensburg@yorku.ca}

\begin{abstract}
Partition function or Fisher zeros play a fundamental role in the theory of phase transitions 
in models in classical statistical mechanics.  In this paper the properties of partition
function zeros in a square lattice self-avoiding walk model of polymer
adsorption are presented.  Some results constraining the distribution of zeros in the
complex plane, based on mathematical results on the distribution of polynomial
zeros, are presented. Numerical results on the 
distribution of zeros are shown, based on approximate enumeration of square
lattice walks using the GAS algorithm.
\end{abstract}

\begin{figure}[b]
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30 40 30 30 40 30 40 40 50 40 50 30 50 20 60 20 60 10 60 0 
70 0 80 0 80 10 80 20 90 20 90 30 100 30 110 30 110 40 
110 50 100 50 100 60 90 60 90 70  /
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70 30 80 30 80 40 70 40 70 50 60 50 50 50 40 50 30 50 
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110 50 100 50 100 60 90 60 90 70  /
\color{black}
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\endpicture
\caption{A self-avoiding walk from the origin in the square lattice.}
\label{figure1}   %%ZXZ[figure1]
\end{figure}

\section{Introduction}
Partition function zeros in the complex plane (also called  \textit{Fisher zeros} \cite{F65}) 
provide a mathematical mechanism which explains phase transitions in models in classical 
statistical mechanics.   In the thermodynamic limit, Fisher zeros accumulate in 
the complex plane in patterns which create an \textit{edge singularity} at the critical point
on the real axis where a phase transition occurs. 

There is a significant literature devoted to the study of partition function
zeros and the edge-singularity in models of lattice spin systems.  These models
include the Ising model \cite{LW98,LW01}, the $q$-state Potts model 
\cite{KC98,KC98a,KC01,K04}, the van der Waal gas and ideal Bose gas \cite{vD13}.
In contrast, far less is known about the partition function zeros
in models of lattice clusters, including models of
the self-avoiding walk (see figure \ref{figure1}).  

Exact enumeration of self-avoiding walks in the square lattice in 
reference \cite{Lee04} to lengths $n=28$ was used to determine partition
function zeros, and to study the relation between the leading partition function zero
and the thermodynamic properties of the model.
Similar work in the cubic lattice is reported in references \cite{LK10,LK12}.
The partition function zeros in a bond fluctuation model of polymer adsorption
was considered in reference \cite{TL14}.  In this last study 
chains in two dimensions of length up to
$1536$ were generated and the distribution of partition function zeros was examined.

In this paper a study similar to that of reference 
\cite{TL14} is reported.  In particular, some theoretical properties of 
partition function zeros in adsorbing lattice walks in the
square lattice are examined, and some numerical data in support
of these properties are presented. 

\section{Adsorbing walks}

Denote the $d$-dimensional hypercubic lattice by $\mathL^d$ and let $c_n$ be 
the number of self-avoiding walks from the origin length $n$ in $\mathL^d$
(see figure \ref{figure1}).
Since $d^n \leq c_n \leq 2d(2d\minus 1)^{n-1}$ in $\mathL^d$, it follows that
$c_n$ grows exponentially with $n$, and it is known that the limit
\begin{equation}
\lim_{n\to\infty} \sfrac{1}{n} \log c_n = \log \mu_d 
\end{equation}
exists \cite{H57}.   The number $\kappa_d = \log \mu_d$ is the \textit{connective
constant} of the lattice $\mathL^d$, and $\mu_d$ is the hypercubic lattice
\textit{growth constant} of $c_n$, namely $c_n = \mu_d^{n+o(n)}$ 
\cite{BH57}. 


The half lattice $\mathL^d_+$ is defined by
\begin{equation}
\mathL^d_+ = 
\{ \vec{v}\in \mathL^d \svv v_d \geq 0 \} ,
\end{equation}
where $\vec{v} = (v_1,v_2,\ldots,v_d)$ is a vertex in $\mathL^d$.
A self-avoiding walk from the origin in the half lattice $\mathL^d_+$ is a
\textit{positive walk}.  The number of positive walks of length $n$ 
is denoted by $c_n^+$.  Naturally, $c_n^+ \leq c_n$.  It is known that
\begin{equation}
\lim_{n\to\infty} \sfrac{1}{n} \log c_n^+ = 
\lim_{n\to\infty} \sfrac{1}{n} \log c_n = \log \mu_d .
\label{eqn3}   %%ZXZ[eqn3]
\end{equation}
See, for example, reference \cite{HTW82}.

The boundary of the half lattice $\mathL^d_+$ is the $(d\minus 1)$-dimensional
hypercubic lattice given by
\begin{equation}
\partial \mathL^d_+ = \{\vec{v}\in\mathL^d_+ \svv \hbox{$v_d = 0$} \} ,
\end{equation}
and it is equivalent to $\mathL^{d-1}$.
A model of \textit{adsorbing (positive) walks} is obtained by weighing vertices of
a positive walk with the weight or \textit{activity} $a$ whenever the walk 
returns to or visits the boundary $\partial \mathL^d_+$.
This assigns a weight $a^v$ to each positive walk making $v$ visits to
$\partial \mathL^d_+$.  Notice that, by convention, the first vertex at
the origin is not a visit. A \textit{positive} or
\textit{adsorbing walk} with visits weighted by $a^v$
is shown in figure \ref{figure3}.

If $c_n^+(v)$ is the number of positive walks of length $n$ and with
$v$ visits in the adsorbing boundary $\partial \mathL^d_+$, then 
the partition function and limiting free energy of adsorbing walks are given by
\begin{equation}
A_n^+(a) = \sum_{v=0}^n c_n^+(v) \, a^v 
\q\hbox{and}\q 
\C{A}^+(a) = \lim_{n\to\infty} \sfrac{1}{n} \log A_n^+(a) .
\label{eqn4}   %%ZXZ[eqn4]
\end{equation}
Existence of the limiting free energy $\C{A}^+(a)$
was shown in reference \cite{HTW82}, and it is
known that $\C{A}^+(a)$ is a non-decreasing convex function of $\log a$.
Moreover, there is a critical point $a_c^+$ in the model, defined by
$\C{A}^+(a) = \log \mu_d$ if $a\leq a_c^+$ and
$\C{A}^+(a) > \log \mu_d$ if $a>a_c^+$.
The critical point $a_c^+$ is the \textit{adsorption critical point}
in the model, and it is a singular point in the free energy $\C{A}^+(a)$.

\begin{figure}[t]
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120 20 120 30 110 30 100 30 100 40 /
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\caption{A positive or adsorbing self-avoiding walk in the half square lattice.
This positive walk steps from the origin (in the boundary of the
half lattice), and each vertex (other than the first vertex at the origin)
where the walk returns to or visits the boundary
is weighted by an activity $a$.}
\label{figure3}  %%ZXZ[figure3]
\end{figure}


\section{Partition function zeros}

Partition function zeros are the (complex valued) roots of 
$A_n^+(a)$.  Since $A_n^+(a)$ is a polynomial of degree $n$, there
are $n$ zeros $\{a_j\}$, and $A_n^+(a)$ factors as 
\begin{equation}
A_n^+(a) = C_n \prod_{j=1}^n (a\minus a_j)
\end{equation}
where $C_n$ is the (positive) coefficient of $a^n$ in equation \Ref{eqn4}
(it depends on $n$).   Notice that $C_n$ is the number of walks in 
$\mathL^{d-1}$, since the coefficient of $a^n$ is the number of
walks with every vertex in $\partial \mathL_+^d$.
Taking logarithms and dividing by $n$, and letting
$n\to\infty$, gives the limiting free energy of 
the model, namely
\begin{equation}
\C{A}^+(a) = \lim_{n\to\infty} \sfrac{1}{n}\log A_n^+(a) 
= \log \mu_{d-1} + \lim_{n\to\infty} \sfrac{1}{n} \sum_{j=1}^n \log (a\minus a_j) ,
\label{eqn8}   %%ZXZ[eqn8]
\end{equation}
in particular since $\log \mu_{d-1} = \lim_{n\to\infty} \sfrac{1}{n} \log C_n$.
Taking derivatives to $\log a$ gives the \textit{energy density} and 
\textit{specific heat} of the adsorbing walk.  

%
%\subsection{The free energy and the distribution of partition function zeros}
%
%Introduce the real finite measures
%\begin{equation}
%\lambda_n E = \sfrac{1}{n} \sum_{a_j\in E} 1
%\label{eqn10q}   %%ZXZ[eqn10q]
%\end{equation}
%on the $\sigma$-algebra of
%plane measure sets $E\subseteq \RealN^2 \simeq \ComplexN$ (in
%other words, plane measure on $\RealN^2$ is extended to
%plane measure on  $\ComplexN$).  Since $\lambda_n\emptyset
%= 0$, $\lambda_n \ComplexN = 1$, and $\lambda_n$ is monotone 
%(since $\lambda_n E \leq \lambda_n F$ if $E\subseteq F$) and subadditive
%(since $\lambda_n E\cup F \leq \lambda_n E + \lambda_n F$), it follows that
%$\lambda_n$ is a probability measure on $\ComplexN$.  Define the limiting set function
%\begin{equation}
%\lambda E = \limsup_{n\to\infty} \lambda_n E .
%\label{eqn10a}   %%ZXZ[eqn10a]
%\end{equation}
%It may be checked that $\lambda \emptyset = 0$ and $\lambda \ComplexN = 1$,
%that $\lambda$ is monotone and subadditive, and so $\lambda$ is a
%probability measure on $\ComplexN$.  
%
%Let $E$ be a plane measure set in $\ComplexN$, and denote the complement of
%$E$ by $\o{E} = \ComplexN\setminus E$.  Then
%\begin{equation}
%\lambda_n E + \lambda_n \o{E} = \lambda_n \ComplexN = 1 .
%\end{equation}
%In other words,
%\begin{equation}
%\liminf_{n\to\infty}\lambda_n E 
%= \liminf_{n\to\infty} (1 - \lambda_n \o{E})
%= 1 - \limsup_{n\to\infty} \lambda_n \o{E}
%= 1 - \lambda \o{E}
%= \lambda E,
%\end{equation}
%since $\lambda_n$ and $\lambda$ are probability measures on $\ComplexN$.
%By equation \Ref{eqn10a} this gives the following lemma.
%
%\begin{lemma} 
%For any plane measure set $E$ in $\ComplexN$,
%$\displaystyle
%\lambda E = \lim_{n\to\infty} \lambda_n E$. \qed
%\label{lemma21}   %%ZXZ[lemma21]
%\end{lemma}
%
%Since the finite measures $\lambda$ and $\lambda_n$ are absolutely
%continuous with respect to plane measure $\alpha$, by the Radon-Nikodym
%theorem there exists unique non-negative measurable functions
%$p$ and $p_n$ such that for any measure set $E$ in the plane,
%\begin{equation}
%\lambda E = \int_E p \, d\alpha ,
%\q\hbox{and}\q
%\lambda_n E = \int_E p_n \, d\alpha.
%\end{equation}
%The measure 
%\begin{equation}
%\beta_n  E = ( \lambda_n - \lambda ) E = \int_E (p_n\minus p)\,d\alpha
%\end{equation}
%is a signed measure on $\ComplexN$.  By lemma \ref{lemma21},
%\begin{equation}
%\lim_{n\to\infty} \beta_n E = \lim_{n\to\infty} (\lambda_n\minus \lambda) E = 0
%\end{equation}
%for any measure set $E$.
%
%The Hahn decomposition of $\RealN^2 \simeq \ComplexN$ by the measure $\beta_n$
%decomposes $\ComplexN$ in a positive set $F_+^{(n)}$ and a negative
%set $F_-^{(n)}$. Define the mutually singular (positive) 
%measures $\beta_n^+$ and $\beta_n^-$ on the Hahn decomposition 
%of $\ComplexN$:
%\begin{equation}
%\beta_n^+E = \beta_n(E\cap F_+^{(n)}),
%\q\hbox{and}\q
%\beta_n^-E = -\beta_n(E\cap F_-^{(n)}) .
%\end{equation}
%Then it follows that $\beta_n = \beta_n^+ - \beta_n^-$ is a Jordan decomposition
%of the signed measure $\beta_n$.
%Since $\lim_{n\to\infty} \beta_n E = 0$ for any measure set, it follows 
%that $\lim_{n\to\infty} \beta_n^+ E = 0$ and $\lim_{n\to\infty} \beta_n^- E = 0$
%for any measure set $E$.
%
%The total variation of the signed measure $\beta_n$ is given by
%\begin{equation}
%|\beta_n|  = \beta_n^+  + \beta_n^- 
%\end{equation}
%and it is a measure on $\ComplexN$.  It follows that
%$\lim_{n\to\infty} |\beta_n| E = 0$ for any measure set $E$.  Notice that
%\begin{eqnarray}
%|\beta_n| E 
%&=&  \beta_n(E\cap F_+^{(n)})  -\beta_n(E\cap F_-^{(n)}) \cr
%&=& \int_{E\cap F_+^{(n)}} (p_n\minus p)\,d\alpha
% + \int_{E\cap F_-^{(n)}} (p\minus p_n)\,d\alpha \cr
%&=& \int_E |p_n \minus p| \, d\alpha .
%\end{eqnarray}
%Since $\lim_{n\to\infty} |\beta_n| E = 0$ for every measure set $E$
%in $\ComplexN$, it follows that for every measure set $E$,
%\begin{equation}
%\lim_{n\to\infty}\int_E |p_n-p|\,d\alpha = 0.
%\label{eqn22A}   %%ZXZ[eqn22A]
%\end{equation}
%This shows that $\lim_{n\to\infty} |p_n \minus p| = 0$ in measure
%in $\ComplexN$. 
%
%(To see this, notice that $p_n \to p$ in the space $L^1(\alpha)$.
%Let $\eps>0$ and define measure sets 
%$F_n = \{ z\in\ComplexN \svv |p_n(z) \minus p(z)|> \eps \}$.  Then
%\begin{equation}
%\int |p_n \minus p| \, d\alpha \geq \int_{F_n} |p_n\minus p|\, d\alpha
%\geq \eps\,\alpha F_n .
%\end{equation}
%This shows that $\lim_{n\to\infty} \alpha F_n = 0$.   This shows
%that $p_n \to p$ in measure (or that there is a subsequence
%$p_{n_j} \to p$ pointwise almost everywhere).  In other words
%$\lim_{n\to\infty} |p_n \minus p| = 0$ in measure.)
%
%These arguments give the following lemma:
%
%\begin{lemma} 
%$\displaystyle \lim_{n\to\infty} p_n = p$ in measure
%in $\ComplexN$.  Thus there is a subsequence
%$\LA p_{n_j} \RA$ such that  $\lim_{j\to\infty} p_{n_j} = p$ pointwise
%almost everywhere in $\ComplexN$. \qed
%\label{lemma22}   %%ZXZ[lemma22]
%\end{lemma}
%
%Since $\lim_{j\to\infty} p_{n_j} = p$ almost everywhere, the Lebesgue 
%Dominated Convergence theorem shows that
%\begin{equation}
%\lambda E = \lim_{n\to\infty} \lambda_n E
%= \lim_{n\to\infty} \int_E p_n \, d\alpha
%= \lim_{j\to\infty} \int_E p_{n_j} \, d\alpha
%= \int_E \lim_{j\to\infty} p_{n_j} \, d\alpha 
%= \int_E p\, d\alpha.
%\end{equation}
%
%The Radon-Nikodym derivatives of $\lambda$ and $\lambda_n$
%to real plane measure $\alpha$ is
%\begin{equation}
%\Sfrac{d\lambda}{d\alpha} = p,
%\q\hbox{and}\q
%\Sfrac{d\lambda_n}{d\alpha} = p_n .
%\label{eqn10b}   %%ZXZ[eqn10b]
%\end{equation} 
%$\lambda_n$ and $\alpha$ are real measures, and so $p_n$ is real-valued,
%non-negative, and $\int_\ComplexN p_n\, d\alpha = 1$.
%By equation \Ref{eqn10q} the measure $\lambda_n$ is normalised
%counting measure on the $\sigma$-algebra of plane measure sets
%in $\ComplexN \simeq \RealN^2$.
%This shows that $p_n$ is the distribution function
%of partition function zeros of $A_n^+(a)$ in $\ComplexN$.   Since $p=\lim_n p_n$
%in measure in $\ComplexN$, $p$ is a real-valued non-negative function with
%$\int_\ComplexN p\, d\alpha = 1$.  This shows that $p$ is the limiting
%distribution function of partition function zeros of $A_n^+(a)$ in the limit $n\to\infty$.
%
%Since $\lambda$ and $\lambda_n$ are finite measures, and $\alpha$ is
%$\sigma$-finite, lemma \ref{lemma22} and
%the Lebesgue Dominated Convergence theorem
%applied to equation \Ref{eqn10} shows that, for $a\in \RealN$,
%\begin{eqnarray}
%\varrho (a) 
%&=& \lim_{n\to\infty} \sfrac{1}{n} \sum_{j=1}^n \log(a\minus a_j) \cr
%&=& \lim_{n\to\infty} \int \log (a\minus z)\, d\lambda_n(z) \cr
%&=& \lim_{n\to\infty} \int \log (a\minus z)\, p_n(z)\, d\alpha(z) \cr
%&=& \lim_{j\to\infty} \int \log (a\minus z)\, p_{n_j}(z)\, d\alpha(z) \cr
%&=& \int \log (a\minus z)\, p(z)\, d\alpha(z)  = \int \log (a\minus z) \, d\lambda (z) .
%\end{eqnarray}
%In other words, in terms of plane measure $\alpha$ and the limiting
%probability distribution function $p(z)$,
%\begin{equation}
%\varrho (a) = \int \log (a\minus z)\, p(z)\, d\alpha (z) ,
%\label{eqn11}   %%ZXZ[eqn11]
%\end{equation}
%where the integration is over the entire complex plane with real plane measure
%$\alpha$.  Since $C = c_n^{(d-1)}$, the following theorem follows from equations \Ref{eqn8} 
%and \Ref{eqn11}.
%
%\begin{theorem}
%The limiting free energy of adsorbing walks is given by
%\[
%\C{A}^+(a) = \log \mu_{d-1} + \varrho (a) = \log\mu_{d-1} 
%+ \int_\ComplexN \log (a\minus z)\, p(z)\,d\alpha(z) 
%\]
%where $a\in\RealN^+$ (the positive real axis) and
%where $p(z)$ is the limiting distribution function of partition function zeros in
%$\ComplexN$. \qed
%\label{eqn12}   %%ZXZ[eqn12]
%\end{theorem}
%
%Since $\C{A}^+(a) = \log \mu_d$ if $a\in (0,a_c^+]$, it follows that
%$\varrho(a) = \log \sfrac{\mu_d}{\mu_{d-1}}$ for $a\in (0,a_c^+]$.
%In addition, $\varrho (a) \simeq \log a$ if $a\in\RealN$ and $a\to\infty$.
%Numerical approximations of $\varrho(a)$ may be made by approximating the distribution
%of zeros in the complex plane and then using 
%$\sfrac{1}{n} \sum_{j=1}^n\log (a\minus a_j)$ as its approximate value.
%
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%\plot -0.663325 0.350000  -0.357071 0.350000 /  \plot 0.357071 0.350000  0.663325 0.350000 /
%\plot -0.600000 0.450000  -0.217945 0.450000 /  \plot 0.217945 0.450000  0.600000 0.450000 /
%\plot -0.509902 0.550000  0.000000 0.550000 /  \plot 0.000000 0.550000  0.509902 0.550000 /
%\plot -0.374166 0.650000  0.000000 0.650000 /  \plot 0.000000 0.650000  0.374166 0.650000 /
%\plot -0.100000 0.750000  0.100000 0.750000 /  \plot 0.000000 0.750000  0.000000 0.750000 /
%\color{red}
%\setplotsymbol ({\LARGE.})
%\circulararc 360 degrees from 0.5 0 center at 0 0 
%\setquadratic
%\plot 0.5 0 0.65 0.1 0.75 0.2 /
%\plot 0.5 0 0.65 -0.1 0.75 -0.2 /
%\setlinear
%\color{Sepia}
%\put {\small$\bullet$} at 0.5 0
%\color{black}
%\put {(b)} at 0 -1.1 
%\normalcolor
%\endpicture
%\caption{(a) A schematic illustration of the distribution of partition function zeros
%in the $a$-plane for small $n$.  (b) In the limit $n\to\infty$ the distribution of partition 
%function zeros approaches a distribution $p(z)$, and the limiting free energy may be computed
%as in theorem \ref{eqn12}. Notice the appearance of an edge-singularity at 
%the critical point $a_c^+$ on the positive real axis in the complex $a$-plane.}
%\label{figure51}  %%ZXZ[figure51]
%\end{figure}
%
%
%Notice the appearance of an edge singularity in figure \ref{figure51}(b) at $a_c^+$
%on the real axis.  At this point the function $\varrho(a)$ is singular.  This shows that 
%$\C{A}^+(a)$ is singular at $a_c^+$, at the critical point where the walk
%adsorbs on $\partial \mathL^d_+$.    That is, the distribution $p(z)$ is
%singular at the point $z=a_c^+$.
%
%In the limit as $|a|\to\infty$, $\log(a\minus z)= \log a - \sfrac{z}{a} + O(a^{-2})$.  This gives
%\begin{equation}
%\C{A}^+(a) = \log \mu_{d-1} + \log a - \sfrac{1}{a} \int_{\ComplexN} z\,p(z)\,d\alpha
% + O(a^{-2}),
%\end{equation}
%proving that $\C{A}^+(a)$ is asymptotic to $\log \mu_{d-1} + \log a$, as also noted
%in the caption of figure \ref{figureF1}. 

\subsection{The distribution of partition function zeros}

The distribution of partition function zeros may be examined by defining the 
functions
\begin{eqnarray}
\nu_n(\rho) &=& \#  \{ \hbox{$\C{A}^+(a)$ zeros $a_k$} \svv (1\minus \rho) \leq |a_k| \leq
\sfrac{1}{1-\rho} \}  \cr
\alpha_n(\theta,\phi) &=& \# \{ \hbox{$\C{A}^+(a)$ zeros $a_k$} \svv
\theta < \Arg a_k \leq \phi \} .
\label{eqn15}  %%ZXZ[eqn15]
\end{eqnarray}
That is, $\nu_n(\rho)$ is the number of zeros of $\C{A}^+(a)$ in the annulus
$(1\minus \rho) \leq |a| \leq \sfrac{1}{1-\rho}$ in the $a$-plane
(where $\rho\in (0,1)$), and $\alpha_n(\theta,\phi)$
is the number of zeros $a_k$ with complex principal arguments $\Arg a_k$ greater 
than $\theta$ and less than or equal to $\phi$.
%
%\begin{figure}[t]
%\beginpicture
%\setcoordinatesystem units <60pt,60pt>
%\setplotarea x from -3.7 to 1.5, y from -1.55 to 1.55
%\setplotarea x from -1.5 to 1.5, y from -1.5 to 1.5
%
%\color{black}
%\normalcolor
%
%\multiput {} at -1 -1 1 1 /
%\scriptsize
%\setplotarea x from -1.5 to 2, y from -1.5 to 1.5
%\axis bottom shiftedto y=0
% ticks withvalues {-$1.5$} {-$1$} {} {} {} {$1$} {$1.5$} {} / quantity 8 /
%%\axis bottom shiftedto y=0
%% label {\hspace{10cm}$\Re a_k$} /
%\axis left shiftedto x=0
% ticks withvalues {-$1.5$i} {-$1$i} {} {} {} {$1$i} {$1.5$i} {} / quantity 7 /
%%\axis left shiftedto x=0
%% label shiftedto y=1  {$\Im a_k$} /
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%
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%
%\setplotsymbol ({\LARGE.})
%
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%\arrow <8pt> [.2,.67] from 0 0 to -0.5 0.5
%\put {\footnotesize$\frac{1}{1\minus\rho}$} at -0.75 1.3
%\arrow <8pt> [.2,.67] from 0 0 to -0.5 1.20
%
%\color{red}
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%
%\setcoordinatesystem units <60pt,60pt> 
%
%\setplotsymbol ({\footnotesize$\bullet$})
%
%\color{Sepia}
%\plot 1.5 1.5 0 0 2 0.6 /
%
%\setplotsymbol ({\LARGE.})
%\color{blue}
%\circulararc 45 degrees from 1.5 0 center at 0 0 
%\put {$\phi$} at 1.4 0.8
%\circulararc 17 degrees from 2 0 center at 0 0 
%\put {$\theta$} at 2.1 0.3
%
%\color{black}
%\multiput {\Large$\bullet$} at 0.3 -0.4 0.6 0.8 0.7 0.9 -0.6 0.7 -0.65 -0.75
%-0.68 -0.9 0.55 -1  0.65 0.6  0.9 0.7 1.0 0.7 0.6 -1.4 -1.1 -.5 -1.3 -0.7 -1.4 1.0 -0.9 0.3  /
%\setlinear
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%\setsolid
%
%\normalcolor
%\endpicture
%\caption{Partition function zeros in $\ComplexN$.
%The distribution of these zeros can be determined by
%counting the number of zeros in an annulus with boundary circles
%$|z|=1\minus\rho$ and $|z| = \frac{1}{1-\rho}$ (where
%$\rho\in(0,1)$, and by counting the zeros inside a half-open 
%wedge $\theta <\Arg z \leq \phi$
%(where $\Arg z$ is the principal argument of $z\in\ComplexN$).
%This gives the functions $\nu_n (\rho)$ and $\alpha_n(\theta,\phi)$
%in equation \Ref{eqn15}.}
%\label{figureV}   %%ZXZ[figureV]
%\end{figure}
%
%Two key general theorems on $\nu_n(\rho)$ and $\alpha_n(\theta,\phi)$ are
%the following:
%
%
%\begin{theorem}[Hughes \& Nikeghbali \cite{HN08}]
%Let $b_k \in \ComplexN$ and assume that $b_0b_n\not=0$.
%Define the polynomial $P(t) = \sum_{j=0}^n b_j\,t^j$.
%Suppose that
%\[ L_n = \log \sum_{j=0}^n |b_j| - \sfrac{1}{2} \log |b_0| - \sfrac{1}{2} \log |b_n| . \] 
%Then, for $0< \rho < 1$,
%\[0\leq 1\minus \sfrac{1}{n}\, \nu_n(\rho) \leq \sfrac{2}{n\rho}\, L_n . \]
%If $L_n= o(n)$, then $\lim_{n\to\infty} \sfrac{1}{n} \nu_n(\rho) = 1$ for
%any $\rho\in (0,1)$. \qed
%\label{thm1}   %%ZXZ{thm1}
%\end{theorem}
%
%\begin{theorem}[Erd\"os \& Tur\'an \cite{ET50}] 
%Let $b_k \in \ComplexN$ and assume that $b_0b_n\not=0$.
%Define the polynomial $P(t) = \sum_{j=0}^n b_j\,t^j$.
%Suppose that
%\[ L_n = \log \sum_{j=0}^n |b_j| - \sfrac{1}{2} \log |b_0| - \sfrac{1}{2} \log |b_n| . \]
%Then
%\[ \LV \sfrac{1}{n}\, \alpha_n(\theta,\phi) - \sfrac{1}{2\pi}(\phi \minus\theta) \RV^2
%\leq \sfrac{C}{n}\, L_n ,\]
%for some non-negative constant $C$. 
%
%If $L_n = o(n)$, then $\displaystyle \lim_{n\to\infty} \sfrac{1}{n}\, \alpha_n(\theta,\phi)
%= \sfrac{1}{2\pi}(\phi\minus \theta)$.
%\qed
%\label{thm2}   %%ZXZ{thm2}
%\end{theorem}

The theorem by Hughes and Nikeghbali \cite{HN08} is useful in providing bounds
on $\nu_n(\rho)$, while the angular distribution of zeros can be examined using
the classical theorem of Erd\"os and Tur\'an \cite{ET50}. Let $\zeta_d 
= \log \sfrac{\mu_d}{\mu_{d-1}}$, then, if $\zeta_d < \rho < 1$, then Hughes
and Nikeghbali's theorem implies that
\begin{equation}
0 < 1- \sfrac{1}{\rho}\,\zeta_d \leq \liminf_{n\to\infty} \sfrac{1}{n}\, \nu_n(\rho) 
\leq \limsup_{n\to\infty} \sfrac{1}{n}\, \nu_n(\rho) \leq 1 .
\label{eqnHN}  %%ZXZ[eqnHN]
\end{equation}
The theorem by Erd\"os and Tur\'an similarly shows that there is a constant $C_0$ so that,
if $\zeta_d < \rho < 1$, then
\begin{equation}
\limsup_{n\to\infty}  \LV \sfrac{1}{2\pi}(\phi\minus\theta) - 
\sfrac{1}{n}\, \alpha_n(\theta,\phi) \RV \leq \sqrt{\sfrac{1}{2}C_0\zeta_d} . 
\label{eqnET}   %%ZXZ[eqnET]
\end{equation}
These results strongly suggest that partition function zeros are distributed in
an annular region in the complex plane, and that the angular distribution is
asymptotically fairly even.



\subsection{Numerical results}
\label{numericalresults}

Numerical data were collected on adsorbing walks using the GAS algorithm
\cite{JvRR09} which was implemented for adsorbing walks in reference \cite{JvR16}.  These
simulations produced estimates of $c_n^+(v)$ for $0\leq v \leq n \leq 500$
in the microcanonical ensemble.  Combining these data as in equation
\Ref{eqn4} produces estimates of the partition function $A_n^+(a)$.  
Zeros of $A_n^+(a)$ were obtained to high accuracy by deflating the
polynomials in Maple 17 \cite{Maple}.   In figure \ref{figureA} the partition function
zeros for adsorbing square lattice walks are shown.

\begin{figure}[t]
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\axis bottom shiftedto y=0
 ticks withvalues {-4} {} {-2} {} {} {} {2} {} {4} / quantity 9 /
%\axis bottom shiftedto y=0
% label {\hspace{10cm}$\Re a_k$} /
\axis left shiftedto x=0
 ticks out withvalues {-6$i$} {} {-4$i$} {} {-2$i$} {} {} {} {2$i$} {} {4$i$} {} {6$i$} / quantity 13 /
%\axis left shiftedto x=0
% label shiftedto y=1  {$\Im a_k$} /
\normalcolor


\endpicture
\caption{Partition function zeros of the partition function
$A_n^+(a)$ of adsorbing self-avoiding square lattice walks.  The data
shown are for walks of length $n\in\{25,50,75,100,125,150\}$.  }
\label{figureA}   %%ZXZ[figureA]
\end{figure}

Finite size approximations to the free energy can be computed from the partition
function zeros using equation \Ref{eqn8}.  Similarly, the energy density $\C{E}^+$
and specific heat $\C{C}^+$ are given by 
\begin{equation}
\C{E}^+(a) 
= \lim_{n\to\infty} \sfrac{1}{n} \sum_{j=1}^n \Sfrac{1}{a\minus a_j},
\q\hbox{and}\q
\C{C}^+(a) 
= \lim_{n\to\infty} \sfrac{-1}{n} \sum_{j=1}^n \Sfrac{1}{(a\minus a_j)^2} .
\end{equation}
In figure \ref{figureB} finite size approximations to the
excess free energy $\C{A}^+(a) \minus \log\mu_d$, energy density
and specific heat,  are plotted for $n\in\{60,125,250,500\}$.  Notice the sharp peak 
in the specific heat curves, and that these curves intersect close to 
a common point at $\log a_c^+$ in the graph. For values $\log a < \log a_c^+$  
the curves decrease with increasing $n$, and
for $\log a>\log a_c^+$ the curves increase with increasing $n$.  This intersection is
therefore a good estimage of the critical adsorption activity in the model.
By calculating it for $(n,2n)$ and $n\in\{100,110,120,\ldots,250\}$ a set
of estimates of $a_c^+$ is obtained (see equation (58) in reference
\cite{JvR16}).  Taking the average to find the best estimate
gives 
\begin{equation}
a_c^+ = 1.764(16),\;\hbox{in $\mathL^2_+$}.
\label{eqn32}  %%ZXZ[eqn32]
\end{equation}
This estimate can be compared to the best estimate 
$a_c^+ = 1.77564$ in the square lattice using exact enumeration methods
\cite{BGJ12} and $a_c^+ = 1.779$ in reference \cite{JvR16}.


\begin{figure}[t!]
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/

\setlinear
\color{black}
\setsolid
\normalcolor

\endpicture
\caption{The excess free energy, energy density and specific heat of
adsorbing walks as a function of $a$ on a log-linear scale for
walks of length $n \in\{60,125,250,500\}$.}
\label{figureB}   %%ZXZ[figureB]
\end{figure}


\subsection{The cumulative angular distribution function}

A \textit{cumulative angular distribution function}
$\hbox{\Large$\alpha$}_n(\phi)$ of partition function zeros (of $A_n^+(a)$)
around the origin in the complex $a$-plane is defined by using the
function $\alpha_n(\theta,\phi)$ defined in equation \Ref{eqn15}:
\begin{equation}
\hbox{\Large$\alpha$}_n(\phi) = \sfrac{1}{n} \alpha_n(-\pi,\phi) .
\label{eqn57}  %%ZXZ[eqn57]
\end{equation}
$\hbox{\Large$\alpha$}_n(\phi)$ is a non-decreasing step-function
which increments in steps of height $\sfrac{1}{n}$ each time an increase in $\phi$
includes new zeros in the interval $(-\pi,\phi]$, and with
$\hbox{\Large$\alpha$}_n(-\pi)=0$ and
$\hbox{\Large$\alpha$}_n(\pi)=1$.  In other words, 
$\hbox{\Large$\alpha$}_n(\phi)$ is the fraction of partition function zeros
with principal argument in the interval $(-\pi,\phi]$.

The cumulative angular distribution function $\hbox{\Large$\alpha$}_n(\phi)$ 
is plotted in figure \ref{figureE} for $n\in\{60, 125, 250, 500 \}$.  It increases with $\phi$,
except near $\phi=0$ where it has a slight flattening (this is for
zeros close to the positive real axis in the $a$-plane).  This
flattening becomes less pronounced as $n$ increases.  In the limit the
data suggest a straight line increasing from $0$ at $\phi=-\pi$ to
$1$ at $\phi=\pi$.  That is, the cumulative distribution function apparently
converges such that $\lim_{n\to\infty}\hbox{\Large$\alpha$}_n(\phi) = 
\sfrac{1}{2\pi}\phi + \sfrac{1}{2}$. This implies that the limiting
angular distribution is given by $\lim_{n\to\infty} \sfrac{1}{n} \alpha_n(\phi,\theta) = 
\sfrac{1}{2\pi}(\phi\minus \theta)$ (this remains unproven).


\section{Conclusions}

In this paper I report on some aspects of partition function zeros in a
self-avoiding walk model of two dimensional adsorbing polymers.  
The data in figure \ref{figureE} suggest strongly that the angular
distribution of zeros, in the limit as $n\to\infty$, is uniform.  
Algebraic results in equations \Ref{eqnHN} and \Ref{eqnET} constraining 
the distribution of partition function zeros were determined, and
tested numerically by examining the distribution of partition function zeros.
It is found that a positive density of zeros is distributed in an annulus with
center at the origin in $\ComplexN$ and that the angular distribution is bounded
in the sense that $\lim_{n\to\infty} \sfrac{1}{n} \alpha_n(\theta,\phi)$ approaches
zero as $\theta\nearrow \phi$ (and the numerical data in figure \ref{figureE} suggest
that it is uniform). 


In section \ref{numericalresults} the partition function zeros are estimated
from numerical data obtained by sampling adsorbing self-avoiding walks
in the square lattice using the GAS algorithm \cite{JvRR09}.  The
free energy is approximated in terms of the partition function zeros, and
in figure \ref{figureB} the excess free energy, energy density and specific heat
are plotted.  The specific heat curves intersect in a point 
approximating the critical adsorption activity $a_c^+$, giving the
estimate in equation \Ref{eqn32}.


\begin{figure}[t]
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%% cumulative distribution function for n=60
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0.55  0.56 0.55  0.57 0.55  0.57 0.55  0.58 0.55  0.58 0.55  0.59 0.55  0.60 0.55  0.60 0.57  0.61 0.57  0.62 0.57  0.62 0.57  0.63 0.57  0.63 0.57  0.64 0.57  0.65 0.57  0.65 0.57  0.66 0.57  0.67 0.57  0.67 0.58  0.68 0.58  0.68 0.58  0.69 0.58  0.70 0.58  0.70 0.58  0.71 0.58  0.72 0.58  0.72 0.58  0.73 0.58  0.74 0.58  0.74 0.58  0.75 0.58  0.75 0.58  0.76 0.58  0.77 0.58  0.77 0.58  0.78 0.58  0.79 0.58  0.79 0.60  0.80 0.60  0.80 0.60  0.81 0.60  0.82 0.60  0.82 0.60  0.83 0.60  0.84 0.60  0.84 0.60  0.85 0.60  0.85 0.60  0.86 0.60  0.87 0.60  0.87 0.60  0.88 0.62  0.89 0.62  0.89 0.62  0.90 0.62  0.90 0.62  0.91 0.62  0.92 0.62  0.92 0.62  0.93 0.62  0.94 0.62  0.94 0.62  0.95 0.63  0.96 0.63  0.96 0.63  0.97 0.63  0.97 0.65  0.98 0.65  0.99 0.65  0.99 0.65  1.00 0.65  1.01 0.65  1.01 0.65  1.02 0.65  1.02 0.65  1.03 0.65  1.04 0.65  1.04 0.65  1.05 0.67  1.06 0.67  1.06 0.67  1.07 0.67  1.07 0.67  1.08 0.67  1.09 0.67  1.09 0.67  1.10 0.67  1.11 0.67  1.11 0.67  1.12 0.67  1.12 0.67  1.13 0.67  1.14 0.67  1.14 0.67  1.15 0.67  1.16 0.67  1.16 0.67  1.17 0.67  1.17 0.67  1.18 0.67  1.19 0.67  1.19 0.67  1.20 0.67  1.21 0.67  1.21 0.67  1.22 0.67  1.23 0.67  1.23 0.67  1.24 0.67  1.24 0.67  1.25 0.67  1.26 0.67  1.26 0.68  1.27 0.68  1.28 0.68  1.28 0.68  1.29 0.68  1.29 0.70  1.30 0.70  1.31 0.70  1.31 0.70  1.32 0.70  1.33 0.70  1.33 0.70  1.34 0.70  1.34 0.70  1.35 0.70  1.36 0.70  1.36 0.70  1.37 0.70  1.38 0.70  1.38 0.70  1.39 0.72  1.39 0.72  1.40 0.72  1.41 0.72  1.41 0.72  1.42 0.72  1.43 0.72  1.43 0.72  1.44 0.72  1.45 0.72  1.45 0.72  1.46 0.72  1.46 0.72  1.47 0.72  1.48 0.72  1.48 0.72  1.49 0.72  1.50 0.72  1.50 0.72  1.51 0.72  1.51 0.73  1.52 0.73  1.53 0.73  1.53 0.75  1.54 0.75  1.55 0.75  1.55 0.75  1.56 0.75  1.56 0.75  1.57 0.75  1.58 0.75  1.58 0.75  1.59 0.75  1.60 0.75  1.60 0.75  1.61 0.77  1.61 0.77  1.62 0.77  1.63 0.77  1.63 0.77  1.64 0.77  1.65 0.77  1.65 0.77  1.66 0.78  1.67 0.78  1.67 0.78  1.68 0.78  1.68 0.78  1.69 0.78  1.70 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0.97  2.85 0.97  2.85 0.97  2.86 0.97  2.87 0.97  2.87 0.97  2.88 0.97  2.88 0.97  2.89 0.97  2.90 0.97  2.90 0.97  2.91 0.97  2.92 0.97  2.92 0.97  2.93 0.97  2.93 0.97  2.94 0.97  2.95 0.97  2.95 0.97  2.96 0.97  2.97 0.97  2.97 0.97  2.98 0.97  2.98 0.97  2.99 0.97  3.00 0.97  3.00 0.97  3.01 0.97  3.02 0.97  3.02 0.98  3.03 0.98  3.03 0.98  3.04 0.98  3.05 0.98  3.05 0.98  3.06 0.98  3.07 0.98  3.07 0.98  3.08 0.98  3.09 0.98  3.09 0.98  3.10 0.98  3.10 0.98  3.11 0.98  3.12 0.98  3.12 0.98  3.13 0.98  3.14 0.98  3.14 0.98
/

%% cumulative distribution function for n=125
\color[cmyk]{0.61,0.0,0.48,0.25}
\put {$\bullet$ - $n=125$} at 3 0.75
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0.86  2.27 0.86  2.28 0.86  2.29 0.86  2.29 0.87  2.30 0.87  2.31 0.87  2.31 0.87  2.32 0.87  2.32 0.87  2.33 0.87  2.34 0.87  2.34 0.87  2.35 0.87  2.36 0.87  2.36 0.87  2.37 0.88  2.38 0.88  2.38 0.88  2.39 0.88  2.39 0.88  2.40 0.88  2.41 0.88  2.41 0.88  2.42 0.88  2.43 0.88  2.43 0.88  2.44 0.89  2.44 0.89  2.45 0.89  2.46 0.89  2.46 0.89  2.47 0.89  2.48 0.89  2.48 0.90  2.49 0.90  2.49 0.90  2.50 0.90  2.51 0.90  2.51 0.90  2.52 0.90  2.53 0.90  2.53 0.90  2.54 0.91  2.54 0.91  2.55 0.91  2.56 0.91  2.56 0.91  2.57 0.91  2.58 0.91  2.58 0.91  2.59 0.91  2.59 0.92  2.60 0.92  2.61 0.92  2.61 0.92  2.62 0.93  2.63 0.93  2.63 0.93  2.64 0.93  2.65 0.93  2.65 0.93  2.66 0.93  2.66 0.93  2.67 0.93  2.68 0.93  2.68 0.93  2.69 0.93  2.70 0.93  2.70 0.93  2.71 0.93  2.71 0.93  2.72 0.93  2.73 0.94  2.73 0.94  2.74 0.94  2.75 0.94  2.75 0.94  2.76 0.94  2.76 0.94  2.77 0.94  2.78 0.94  2.78 0.94  2.79 0.94  2.80 0.94  2.80 0.94  2.81 0.94  2.81 0.95  2.82 0.96  2.83 0.96  2.83 0.96  2.84 0.96  2.85 0.96  2.85 0.96  2.86 0.96  2.87 0.96  2.87 0.96  2.88 0.96  2.88 0.96  2.89 0.96  2.90 0.96  2.90 0.96  2.91 0.96  2.92 0.96  2.92 0.96  2.93 0.96  2.93 0.96  2.94 0.96  2.95 0.96  2.95 0.97  2.96 0.97  2.97 0.97  2.97 0.97  2.98 0.97  2.98 0.97  2.99 0.97  3.00 0.97  3.00 0.97  3.01 0.97  3.02 0.97  3.02 0.97  3.03 0.98  3.03 0.98  3.04 0.98  3.05 0.98  3.05 0.98  3.06 0.98  3.07 0.98  3.07 0.98  3.08 0.99  3.09 0.99  3.09 0.99  3.10 0.99  3.10 0.99  3.11 0.99  3.12 0.99  3.12 0.99  3.13 0.99  3.14 0.99  3.14 0.99  
/


%% cumulative distribution function for n=250
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0.95  2.85 0.95  2.85 0.95  2.86 0.95  2.87 0.95  2.87 0.95  2.88 0.95  2.88 0.95  2.89 0.95  2.90 0.95  2.90 0.96  2.91 0.96  2.92 0.96  2.92 0.96  2.93 0.96  2.93 0.96  2.94 0.96  2.95 0.96  2.95 0.96  2.96 0.96  2.97 0.97  2.97 0.97  2.98 0.97  2.98 0.97  2.99 0.97  3.00 0.97  3.00 0.97  3.01 0.97  3.02 0.97  3.02 0.97  3.03 0.97  3.03 0.98  3.04 0.98  3.05 0.98  3.05 0.98  3.06 0.98  3.07 0.98  3.07 0.98  3.08 0.98  3.09 0.98  3.09 0.98  3.10 0.98  3.10 0.98  3.11 0.98  3.12 0.98  3.12 0.98  3.13 0.98  3.14 0.98  3.14 0.98
/

%% cumulative distribution function for n=500
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/

\setlinear
\color{black}
\setsolid

\normalcolor

\color{black}
\setplotarea x from -3.14 to 3.14, y from 0 to 1.0
\axis bottom  ticks
          withvalues {$-\pi$} {0} {$\pi$} / quantity 3 /
          %logged 
          %numbered at 1.6 1.8 2.0 2.2 / 
         % length <0pt> from 1.5 to 4.5 by 0.5
          %from 1.25 to 2.75 by 0.50 
          %/
\axis bottom label {\hspace{7cm} $\phi$} /

\axis left shiftedto x=0
 ticks out withvalues {} {} {$\frac{1}{2}$} {} {1} / quantity 5 /

\normalcolor

\endpicture
\caption{The cumulative distribution function $\frac{1}{n} 
\alpha_n(-\pi,\phi)$ of the principal arguments of partition function 
zeros in the complex plane for finite values of $n$.
These data are for $n\in\{60,125,250,500\}$ and appear to converge
to the line $\frac{1}{2\pi}\phi+\frac{1}{2}$ as $n\to\infty$.}
\label{figureE}  %%ZXZ[figureE]
\end{figure}



\vspace{0.1cm}
\section*{Acknowledgements}
EJJvR acknowledges support from NSERC(Canada) in the form of a Discovery Grant.
\vspace{0.1cm}

\section*{References}
\vspace{0.3cm}
\bibliographystyle{plain}
\bibliography{SAIP-PartZero}


\end{document}







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as those loading the AMS extension fonts 
\verb"msam" and \verb"msbm" (these fonts provide the 
blackboard bold alphabet and various extra maths symbols as well as 
symbols useful in figure captions); an extra style file \verb"iopams.sty" is 
provided to load these packages and provide extra definitions for bold Greek letters. 
\subsection{Headers, footers and page numbers}
Authors should {\it not} add headers, footers or page numbers to the pages of their article---they will
be added by \iopp\ as part of the production process.

\subsection{{\cls\ }package options}
The \cls\ class file has two options `a4paper' and `letterpaper':
\begin{verbatim}
\documentclass[a4paper]{jpconf}
\end{verbatim}

or \begin{verbatim}
\documentclass[letterpaper]{jpconf}
\end{verbatim}

\begin{center}
\begin{table}[h]
\caption{\label{opt}\cls\ class file options.}
%\footnotesize\rm
\centering
\begin{tabular}{@{}*{7}{l}}
\br
Option&Description\\
\mr
\verb"a4paper"&Set the paper size and margins for A4 paper.\\
\verb"letterpaper"&Set the paper size and margins for US letter paper.\\
\br
\end{tabular}
\end{table}
\end{center}

The default paper size is A4 (i.e., the default option is {\tt a4paper}) but this can be changed to Letter by 
using \verb"\documentclass[letterpaper]{jpconf}". It is essential that you do not put macros into the text which alter the page dimensions.

\section{The title, authors, addresses and abstract} 
The code for setting the title page information is slightly different from
the normal default in \LaTeX\ but please follow these instructions as carefully as possible so all articles within a conference have the same style to the title page. 
The title is set in bold unjustified type using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter of the title should be capitalized with the rest in lower case. 
The next information required is the list of all authors' names followed by 
the affiliations. For the authors' names type \verb"\author{#1}", 
where \verb"#1" is the 
list of all authors' names. The style for the names is initials then
surname, with a comma after all but the last 
two names, which are separated by `and'. Initials should {\it not} have 
full stops. First names may be used if desired. The command \verb"\maketitle" is not
required.

The addresses of the authors' affiliations follow the list of authors. 
Each address should be set by using
\verb"\address{#1}" with the address as the single parameter in braces. 
If there is more 
than one address then a superscripted number, followed by a space, should come at the start of
each address. In this case each author should also have a superscripted number or numbers following their name to indicate which address is the appropriate one for them.
 
Please also provide e-mail addresses for any or all of the authors using an \verb"\ead{#1}" command after the last address. \verb"\ead{#1}" provides the text Email: so \verb"#1" is just the e-mail address or a list of emails.  

The abstract follows the addresses and
should give readers concise information about the content 
of the article and should not normally exceed 200 
words. {\bf All articles must include an abstract}. To indicate the start 
of the abstract type \verb"\begin{abstract}" followed by the text of the 
abstract.  The abstract should normally be restricted 
to a single paragraph and is terminated by the command
\verb"\end{abstract}"

\subsection{Sample coding for the start of an article}
\label{startsample}
The code for the start of a title page of a typical paper might read:
\begin{verbatim}
\title{The anomalous magnetic moment of the 
neutrino and its relation to the solar neutrino problem}

\author{P J Smith$^1$, T M Collins$^2$, 
R J Jones$^{3,}$\footnote[4]{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road, 
Bristol BS8 1TS, UK.} and Janet Williams$^3$}

\address{$^1$ Mathematics Faculty, Open University, 
Milton Keynes MK7~6AA, UK}
\address{$^2$ Department of Mathematics, 
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{$^3$ Department of Computer Science, 
University College London, Gower Street, London WC1E~6BT, UK}

\ead{williams@ucl.ac.uk}

\begin{abstract}
The abstract appears here.
\end{abstract}
\end{verbatim}

\section{The text}
The text of the article should should be produced using standard \LaTeX\ formatting. Articles may be divided into sections and subsections, but the length limit provided by the \corg\ should be adhered to.

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from 
colleagues, special work by technical staff or financial support from 
organizations should do so in an unnumbered Acknowledgments section 
immediately following the last numbered section of the paper. The 
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section.

\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts 
the flow of the article, may be consigned to an appendix. 
Any appendices should be included at the end of the main text of the paper, after the acknowledgments section (if any) but before the reference list.
If there are two or more appendices they will be called Appendix A, Appendix B, etc. 
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"\appendix" is used to signify the start of the
appendixes. Thereafter \verb"\section", \verb"\subsection", etc, will 
give headings appropriate for an appendix. To obtain a simple heading of 
`Appendix' use the code \verb"\section*{Appendix}". If it contains
numbered equations, figures or tables the command \verb"\appendix" should
precede it and \verb"\setcounter{section}{1}" must follow it. 

\section{References}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the online version of \jpcs\ references will be linked to their original source or to the article within a secondary service such as INSPEC or ChemPort wherever possible. To facilitate this linking extra care should be taken when preparing reference lists. 

Two different styles of referencing are in common use: the Harvard alphabetical system and the Vancouver numerical system.  For \jpcs, the Vancouver numerical system is preferred but authors should use the Harvard alphabetical system if they wish to do so. In the numerical system references are numbered sequentially throughout the text within square brackets, like this [2], and one number can be used to designate several references.  

\subsection{Using \BibTeX}
We highly recommend the {\ttfamily\textbf\selectfont iopart-num} \BibTeX\ package by Mark~A~Caprio \cite{iopartnum}, which is included with this documentation.

\subsection{Reference lists}
A complete reference should provide the reader with enough information to locate the article concerned, whether published in print or electronic form, and should, depending on the type of reference, consist of:  

\begin{itemize}
\item name(s) and initials;
\item date published;
\item title of journal, book or other publication; 
\item titles of journal articles may also be included (optional);
\item volume number;
\item editors, if any;
\item town of publication and publisher in parentheses for {\it books};
\item the page numbers.
\end{itemize}

Up to ten authors may be given in a particular reference; where 
there are more than ten only the first should be given followed by 
`{\it et al}'. If an author is unsure of a particular journal's abbreviated title it is best to leave the title in 
full. The terms {\it loc.\ cit.\ }and {\it ibid.\ }should not be used. 
Unpublished conferences and reports should generally not be included 
in the reference list and articles in the course of publication should 
be entered only if the journal of publication is known. 
A thesis submitted for a higher degree may be included 
in the reference list if it has not been superseded by a published 
paper and is available through a library; sufficient information 
should be given for it to be traced readily.

\subsection{Formatting reference lists}
Numeric reference lists should contain the references within an unnumbered section (such as \verb"\section*{References}"). The 
reference list itself is started by the code 
\verb"\begin{thebibliography}{<num>}", where \verb"<num>" is the largest
number in the reference list and is completed by
\verb"\end{thebibliography}". 
Each reference starts with \verb"\bibitem{<label>}", where `label' is the label used for cross-referencing. Each \verb"\bibitem" should only contain a reference to a single article (or a single article and a preprint reference to the same article).  When one number actually covers a group of two or more references to different articles, \verb"\nonum"
should replace \verb"\bibitem{<label>}" at
the start of each reference in the group after the first.

For an alphabetic reference list use \verb"\begin{thereferences}" ... \verb"\end{thereferences}" instead of the
`thebibliography' environment and each reference can be start with just \verb"\item" instead of \verb"\bibitem{label}"
as cross referencing is less useful for alphabetic references.

\subsection {References to printed journal articles}
A normal reference to a journal article contains three changes of font (see table \ref{jfonts}) and is constructed as follows:

\begin{itemize}
\item the authors should be in the form surname (with only the first letter capitalized) followed by the initials with no periods after the initials. Authors should be separated by a comma except for the last two which should be separated by `and' with no comma preceding it;
\item the article title (if given) should be in lower case letters, except for an initial capital, and should follow the date;
\item the journal title is in italic and is abbreviated. If a journal has several parts denoted by different letters the part letter should be inserted after the journal in Roman type, e.g. {\it Phys. Rev.} A;
\item the volume number should be in bold type;
\item both the initial and final page numbers should be given where possible. The final page number should be in the shortest possible form and separated from the initial page number by an en rule `-- ', e.g. 1203--14, i.e. the numbers `12' are not repeated.
\end{itemize}

A typical (numerical) reference list might begin

\medskip
\begin{thebibliography}{9}
\item Strite S and Morkoc H 1992 {\it J. Vac. Sci. Technol.} B {\bf 10} 1237 
\item Jain S C, Willander M, Narayan J and van Overstraeten R 2000 
{\it J. Appl. Phys}. {\bf 87} 965 
\item Nakamura S, Senoh M, Nagahama S, Iwase N, Yamada T, Matsushita T, Kiyoku H 
and 	Sugimoto Y 1996 {\it Japan. J. Appl. Phys.} {\bf 35} L74 
\item Akasaki I, Sota S, Sakai H, Tanaka T, Koike M and Amano H 1996 
{\it Electron. Lett.} {\bf 32} 1105 
\item O'Leary S K, Foutz B E, Shur M S, Bhapkar U V and Eastman L F 1998 
{\it J. Appl. Phys.} {\bf 83} 826 
\item Jenkins D W and Dow J D 1989 {\it Phys. Rev.} B {\bf 39} 3317 
\end{thebibliography}
\smallskip

\noindent which would be obtained by typing

\begin{verbatim}
\begin{\thebibliography}{9}
\item Strite S and Morkoc H 1992 {\it J. Vac. Sci. Technol.} B {\bf 10} 1237 
\item Jain S C, Willander M, Narayan J and van Overstraeten R 2000 
{\it J. Appl. Phys}. {\bf 87} 965 
\item Nakamura S, Senoh M, Nagahama S, Iwase N, Yamada T, Matsushita T, Kiyoku H 
and 	Sugimoto Y 1996 {\it Japan. J. Appl. Phys.} {\bf 35} L74 
\item Akasaki I, Sota S, Sakai H, Tanaka T, Koike M and Amano H 1996 
{\it Electron. Lett.} {\bf 32} 1105 
\item O'Leary S K, Foutz B E, Shur M S, Bhapkar U V and Eastman L F 1998 
{\it J. Appl. Phys.} {\bf 83} 826 
\item Jenkins D W and Dow J D 1989 {\it Phys. Rev.} B {\bf 39} 3317 
\end{\thebibliography}
\end{verbatim}

\begin{center}
\begin{table}[h]
\centering
\caption{\label{jfonts}Font styles for a reference to a journal article.} 
\begin{tabular}{@{}l*{15}{l}}
\br
Element&Style\\
\mr
Authors&Roman type\\
Date&Roman type\\
Article title (optional)&Roman type\\
Journal title&Italic type\\
Volume number&Bold type\\
Page numbers&Roman type\\
\br
\end{tabular}
\end{table}
\end{center}

\subsection{References to \jpcs\ articles}
Each conference proceeding published in \jpcs\ will be a separate {\it volume}; 
references should follow the style for conventional printed journals. For example:\vspace{6pt}
\numrefs{1}
\item Douglas G 2004 \textit{J. Phys.: Conf. Series} \textbf{1} 23--36
\endnumrefs

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{References to preprints}
For preprints there are two distinct cases:
\renewcommand{\theenumi}{\arabic{enumi}}
\begin{enumerate}
\item Where the article has been published in a journal and the preprint is supplementary reference information. In this case it should be presented as:
\medskip
\numrefs{1}
\item Kunze K 2003 T-duality and Penrose limits of spatially homogeneous and inhomogeneous cosmologies {\it Phys. Rev.} D {\bf 68} 063517 ({\it Preprint} gr-qc/0303038)
\endnumrefs
\item Where the only reference available is the preprint. In this case it should be presented as
\medskip
\numrefs{1}
\item Milson R, Coley A, Pravda V and Pravdova A 2004 Alignment and algebraically special tensors {\it Preprint} gr-qc/0401010
\endnumrefs
\end{enumerate}

\subsection{References to electronic-only journals}
In general article numbers are given, and no page ranges, as most electronic-only journals start each article on page 1.

\begin{itemize} 
\item For {\it New Journal of Physics} (article number may have from one to three digits)
\numrefs{1}
\item Fischer R 2004 Bayesian group analysis of plasma-enhanced chemical vapour deposition data {\it New. J. Phys.} {\bf 6} 25 
\endnumrefs
\item For SISSA journals the volume is divided into monthly issues and these form part of the article number

\numrefs{2}
\item Horowitz G T and Maldacena J 2004 The black hole final state {\it J. High Energy Phys.}  	JHEP02(2004)008
\item Bentivegna E, Bonanno A and Reuter M 2004 Confronting the IR fixed point cosmology 	with 	high-redshift observations {\it J. Cosmol. Astropart. Phys.} JCAP01(2004)001  
\endnumrefs
\end{itemize} 

\subsection{References to books, conference proceedings and reports}
References to books, proceedings and reports are similar to journal references, but have 
only two changes of font (see table~\ref{book}). 

\begin{table}
\centering
\caption{\label{book}Font styles for references to books, conference proceedings and reports.}
\begin{tabular}{@{}l*{15}{l}}
\br
Element&Style\\
\mr
Authors&Roman type\\
Date&Roman type\\
Book title (optional)&Italic type\\
Editors&Roman type\\
Place (city, town etc) of publication&Roman type\\
Publisher&Roman type\\
Volume&Roman type\\
Page numbers&Roman type\\
\br
\end{tabular}
\end{table}

Points to note are:
\medskip
\begin{itemize}
\item Book titles are in italic and should be spelt out in full with initial capital letters for all except minor words. Words such as Proceedings, Symposium, International, Conference, Second, etc should be abbreviated to {\it Proc.}, {\it Symp.}, {\it Int.}, {\it Conf.}, {\it 2nd}, respectively, but the rest of the title should be given in full, followed by the date of the conference and the town or city where the conference was held. For Laboratory Reports the Laboratory should be spelt out wherever possible, e.g. {\it Argonne National Laboratory Report}.
\item The volume number, for example vol 2, should be followed by the editors, if any, in a form such as `ed A J Smith and P R Jones'. Use {\it et al} if there are more than two editors. Next comes the town of publication and publisher, within brackets and separated by a colon, and finally the page numbers preceded by p if only one number is given or pp if both the initial and final numbers are given.
\end{itemize}

Examples taken from published papers:
\medskip

\numrefs{99}
\item Kurata M 1982 {\it Numerical Analysis for Semiconductor Devices} (Lexington, MA: Heath)
\item Selberherr S 1984 {\it Analysis and Simulation of Semiconductor Devices} (Berlin: Springer)
\item Sze S M 1969 {\it Physics of Semiconductor Devices} (New York: Wiley-Interscience)
\item Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} (Moscow: Moscow State University Press) p 103
\item Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. on Nuclear Physics (Munich)} vol 1 (Amsterdam: 	North-Holland/American Elsevier) p 517
\item Cheng G X 2001 {\it Raman and Brillouin Scattering-Principles and Applications} (Beijing: Scientific) 
\item Szytula A and Leciejewicz J 1989 {\it Handbook on the Physics and Chemistry of Rare Earths} vol 12, ed K A Gschneidner Jr and L Erwin (Amsterdam: Elsevier) p 133
\item Kuhn T 1998 {\it Density matrix theory of coherent ultrafast dynamics Theory of Transport Properties of Semiconductor Nanostructures} (Electronic Materials vol 4) ed E Sch\"oll (London: Chapman and Hall) chapter 6 pp 173--214
\endnumrefs

\section{Tables and table captions}
Tables should be numbered serially and referred to in the text 
by number (table 1, etc, {\bf rather than} tab. 1). Each table should be a float and be positioned within the text at the most convenient place near to where it is first mentioned in the text. It should have an 
explanatory caption which should be as concise as possible. 

\subsection{The basic table format}
The standard form for a table is:
\begin{verbatim}
\begin{table}
\caption{\label{label}Table caption.}
\begin{center}
\begin{tabular}{llll}
\br
Head 1&Head 2&Head 3&Head 4\\
\mr
1.1&1.2&1.3&1.4\\
2.1&2.2&2.3&2.4\\
\br
\end{tabular}
\end{center}
\end{table}
\end{verbatim}

The above code produces table~\ref{ex}.

\begin{table}[h]
\caption{\label{ex}Table caption.}
\begin{center}
\begin{tabular}{llll}
\br
Head 1&Head 2&Head 3&Head 4\\
\mr
1.1&1.2&1.3&1.4\\
2.1&2.2&2.3&2.4\\
\br
\end{tabular}
\end{center}
\end{table}

Points to note are:
\medskip
\begin{enumerate}
\item The caption comes before the table.
\item The normal style is for tables to be centred in the same way as
equations. This is accomplished
by using \verb"\begin{center}" \dots\ \verb"\end{center}".

\item The default alignment of columns should be aligned left.

\item Tables should have only horizontal rules and no vertical ones. The rules at
the top and bottom are thicker than internal rules and are set with
\verb"\br" (bold rule). 
The rule separating the headings from the entries is set with
\verb"\mr" (medium rule). These commands do not need a following double backslash.

\item Numbers in columns should be aligned as appropriate, usually on the decimal point;
to help do this a control sequence \verb"\lineup" has been defined 
which sets \verb"\0" equal to a space the size of a digit, \verb"\m"
to be a space the width of a minus sign, and \verb"\-" to be a left
overlapping minus sign. \verb"\-" is for use in text mode while the other
two commands may be used in maths or text.
(\verb"\lineup" should only be used within a table
environment after the caption so that \verb"\-" has its normal meaning
elsewhere.) See table~\ref{tabone} for an example of a table where
\verb"\lineup" has been used.
\end{enumerate}

\begin{table}[h]
\caption{\label{tabone}A simple example produced using the standard table commands 
and $\backslash${\tt lineup} to assist in aligning columns on the 
decimal point. The width of the 
table and rules is set automatically by the 
preamble.} 

\begin{center}
\lineup
\begin{tabular}{*{7}{l}}
\br                              
$\0\0A$&$B$&$C$&\m$D$&\m$E$&$F$&$\0G$\cr 
\mr
\0\023.5&60  &0.53&$-20.2$&$-0.22$ &\01.7&\014.5\cr
\0\039.7&\-60&0.74&$-51.9$&$-0.208$&47.2 &146\cr 
\0123.7 &\00 &0.75&$-57.2$&\m---   &---  &---\cr 
3241.56 &60  &0.60&$-48.1$&$-0.29$ &41   &\015\cr 
\br
\end{tabular}
\end{center}
\end{table}
 
\section{Figures and figure captions}
Figures must be included in the source code of an article at the appropriate place in the text not grouped together at the end. 

Each figure should have a brief caption describing it and, if 
necessary, interpreting the various lines and symbols on the figure. 
As much lettering as possible should be removed from the figure itself and 
included in the caption. If a figure has parts, these should be 
labelled ($a$), ($b$), ($c$), etc. 
\Tref{blobs} gives the definitions for describing symbols and lines often
used within figure captions (more symbols are available
when using the optional packages loading the AMS extension fonts).

\begin{table}[h]
\caption{\label{blobs}Control sequences to describe lines and symbols in figure 
captions.}
\begin{center}
\begin{tabular}{lllll}
\br
Control sequence&Output&&Control sequence&Output\\
\mr
\verb"\dotted"&\dotted        &&\verb"\opencircle"&\opencircle\\
\verb"\dashed"&\dashed        &&\verb"\opentriangle"&\opentriangle\\
\verb"\broken"&\broken&&\verb"\opentriangledown"&\opentriangledown\\
\verb"\longbroken"&\longbroken&&\verb"\fullsquare"&\fullsquare\\
\verb"\chain"&\chain          &&\verb"\opensquare"&\opensquare\\
\verb"\dashddot"&\dashddot    &&\verb"\fullcircle"&\fullcircle\\
\verb"\full"&\full            &&\verb"\opendiamond"&\opendiamond\\
\br
\end{tabular}
\end{center}
\end{table}


Authors should try and use the space allocated to them as economically as possible. At times it may be convenient to put two figures side by side or the caption at the side of a figure. To put figures side by side, within a figure environment, put each figure and its caption into a minipage with an appropriate width (e.g. 3in or 18pc if the figures are of equal size) and then separate the figures slightly by adding some horizontal space between the two minipages (e.g. \verb"\hspace{.2in}" or \verb"\hspace{1.5pc}". To get the caption at the side of the figure add the small horizontal space after the \verb"\includegraphics" command and then put the \verb"\caption" within a minipage of the appropriate width aligned bottom, i.e. \verb"\begin{minipage}[b]{3in}" etc (see code in this file used to generate figures 1--3).

Note that it may be necessary to adjust the size of the figures (using optional arguments to \verb"\includegraphics", for instance \verb"[width=3in]") to get you article to fit within your page allowance or to obtain good page breaks.

\begin{figure}[h]
\begin{minipage}{14pc}
\includegraphics[width=14pc]{name.jpg}
\caption{\label{label}Figure caption for first of two sided figures.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{14pc}
\includegraphics[width=14pc]{name.jpg}
\caption{\label{label}Figure caption for second of two sided figures.}
\end{minipage} 
\end{figure}

\begin{figure}[h]
\includegraphics[width=14pc]{name.jpg}\hspace{2pc}%
\begin{minipage}[b]{14pc}\caption{\label{label}Figure caption for a narrow figure where the caption is put at the side of the figure.}
\end{minipage}
\end{figure}

Using the graphicx package figures can be included using code such as:
\begin{verbatim}
\begin{figure}
\begin{center}
\includegraphics{name.eps}
\end{center}
\caption{\label{label}Figure caption}
\end{figure}
\end{verbatim}

\section*{References}

\vspace{1cm}
\noindent{\bf References}
\bibliographystyle{plain}
\bibliography{SAIP-PartZero}




%\begin{thebibliography}{9}
%\bibitem{iopartnum} IOP Publishing is to grateful Mark A Caprio, Center for Theoretical Physics, Yale University, for permission to include the {\tt iopart-num} \BibTeX package (version 2.0, December 21, 2006) with  this documentation. Updates and new releases of {\tt iopart-num} can be found on \verb"www.ctan.org" (CTAN). 
%\end{thebibliography}

\end{document}


