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\begin{document}
\title{VIGRE Application Part II $\cdot$ Spring 2010}
\author{John Kerl}
%\date{\today}
\date{November 4, 2009}
\maketitle

%% ----------------------------------------------------------------
\section{Timeline, professional development, and outreach activities}

I am in my final year of a five-year PhD program.  The period of support
(spring 2010) will complete my fifth year.  The summer and fall of 2009, which
have been on VIGRE support, have seen the following:

\begin{itemize}

\item I have continued my research.  This is discussed in detail below.

\item In June, UA's Bob Sims hosted a workshop on quantum spin systems and
applications in quantum computation.  I gave a talk on lattice quadrupling for
percolation in quantum networks, as described in my summer/fall 2009 VIGRE
proposal:  I am doing follow-on work to \cite{bib:entdist} for 3D rectangular
lattices.  In this talk, I brought my written presentation up to my current
level of understanding.  The only remaining obstacle to publication of this
percolation work is final write-up along with finite-size scaling analysis,
which is hand-in-hand with FSS as applied to my dissertation research.

\item In July, I delivered a contributed talk at the Conference on Stochastic
Processes and Their Applications in Berlin.  My aim was to pitch the model to
probabilists, downplaying the mathematical-physics background and emphasizing
intuition, known results, conjectures, and experimental methods.  While in
Berlin, I made several contacts and had several productive conversations,
including a possible postdoctoral research idea involving minimal-distance
matching of Poisson point processes.

\item Wednesday through Friday of the week before the Berlin conference, I
visited Daniel Ueltschi and his collaborators Daniel Gandolfo and Jean Ruiz in
Marseille.  I received valuable feedback on my slides for my then-upcoming
Berlin talk.  Gandolfo and I compared several technical details for
simulational methods for the model of random spatial permutations, including
size reduction for lookup tables and the irrelevance of Binder fourth-order
cumulants for our model.

\item  Friday August 7 through Tuesday August 11, I participated in the
department's integration workshop for incoming graduate students.  I wrote a
project on the Berlekamp algorithm for factorization of polynomials over finite
fields, an old area of expertise from my master's-degree days.  None of the
eight incoming students chose to work on that project; I assisted Angel Chavez
and Kevin Davidsaver on their group-representation project, focusing in
particular on presentation skills.

\item I have been working with Tom Kennedy's bridge group, which consists of
three graduate students, two undergraduates, and himself.  I am focusing on
mentoring of an undergraduate student (Howard Cheng) and a junior graduate
student (Shane Passon) on software implementation of probabilistic concepts.

\item I presented the progress of my research to the UA mathematical physics
seminar in September.  This included most of the content of my 30-minute Berlin
talk, as well as recently completed work on sample variance of exponentially
correlated Markov processes.  (Precisely, this uses integrated autocorrelation
time to estimate error bars for output of Markov chain Monte Carlo simulation
runs.) I treated this as an early dry run of a job talk; I received several
useful feedback points which will positively impact my on-site job talks this
coming spring.

\item On October 26, I gave a talk to the UA Math Department's Software
Interest Group on the use of the Python programming language for numerical
methods and presentation graphics.  This melded a talk on the Perl language
from three years ago with an explanation of how I use Python's \texttt{pylab}
module to create the figures in my dissertation and associated writings
(including figure \ref{fig:fI_fS} of this proposal).

\item I am completing my final course --- Probability and Random Processes in
Engineering, in the Electrical Engineering department --- satisfying an
out-of-department requirement.

\item Since the start of summer 2009, I have completed the following chapters
of my dissertation: definitions and intuition for the probability model, random
variables of interest in the model, correctness of the GKU algorithm,
correctness of the worm algorithm and discussion of its remaining stopping-time
problem, $\Delta H$ computations, and software design.  I also completed an
appendix on sample variance of exponentially correlated Markov processes.

\item At Daniel Ueltschi's request, I am drafting a paper stating my
dissertation results.  On Tom Kennedy's advice, I have structured this so that
it will drop neatly into my dissertation.  In particular, it will complete my
dissertation chapters on finite-size scaling and quantitative results.  I plan
to submit this paper by the end of the year for publication:  this will enhance
my job-application materials.

\item Since early September I have been actively job-seeking for academic,
governmental/laboratory, and industrial positions.  At present (early
November), I have been gratified to receive two phone interviews and an on-site
interview.

\end{itemize}

The period of support, spring 2010, will include the following:

\begin{itemize}

\item Principally, I will complete my doctoral research.  This is discussed in
detail below.

\item Finite-size scaling analysis, in progress now, is not only a key step for
the quantitative results in my dissertation.  It is also the missing piece for
proper write-up of the percolation results described in my summer/fall 2009
proposal, which I spoke on in June.

\item I will continue to work with Tom Kennedy's bridge group.

\item I will contribute a talk at the AMS/MAA Joint Meetings in San Francisco
in January.  Of course, my primary purpose for attending this year's meetings
will be the job search.

\item I will also attend the 23rd Annual Workshop on Recent Developments in
Computer Simulation Studies in Condensed Matter Physics at the end of February
at the University of Georgia; I am submitting a request for a contributed talk.
There, I hope to make professional contacts; as well, I hope to obtain on
advice on the stopping-time problem of my worm algorithm for the model
of random spatial permutations.

\item The results section of my dissertation will be all but complete by the
beginning of the spring semester, thanks to Daniel's draft-paper request as
described above; I will drop in final numbers once the requisite
high-performance-computing runs have completed.  I will also tie up all loose
ends in my dissertation.

\item Spring job-seeking efforts will include attendance at the Joint Meetings,
possible on-site interviews, and (one fervently hopes) selection from among
multiple job offers.

\item I plan to defend my dissertation at the end of March.  This should leave
time for any re-work, in time for my graduation in May.  In particular,
Daniel Ueltschi will be on campus at that time.

\end{itemize}

VIGRE support will be particularly helpful in permitting me to focus on
completing a quality dissertation which will facilitate my success on the job
market.

%% ----------------------------------------------------------------
\section{Plan of study and research}

My research is under Daniel Ueltschi, formerly of the University of Arizona,
currently at the University of Warwick.  We are studying the effects of
interparticle interactions on the critical temperature of Bose-Einstein
condensation.  Ueltschi spent a sabbatical semester at the UA for spring 2009,
during which time we met regularly; at present, I am working largely
independently.  We will continue to communicate; as well, I am working with my
local advisor, Tom Kennedy, to bring my dissertation to completion.  Daniel
will visit the University of Arizona for a few weeks in March; in particular, I
will schedule my dissertation defense around his visit.

%% ----------------------------------------------------------------
\subsection{Background}

The model of random spatial permutations arises in the study of the Bose gas,
although it is also of intrinsic probabilistic interest; its history includes
Bose-Einstein \cite{bib:bose,bib:einstein}, Feynman\cite{bib:feynman},
Penrose-Onsager, S\"ut\H{o} \cite{bib:suto1,bib:suto2}, and Ueltschi-Betz
\cite{bib:bu,bib:qmath,bib:bu07,bib:bu08}.  Random permutations arise
physically when one symmetrizes the $N$-boson Hamiltonian with pair
interactions, then applies a multi-particle Feynman-Kac formula and a cluster
expansion \cite{bib:bu07,bib:bu08}.

The state space is $\Omega_{\Lambda,N} = \Lambda^N \times \SN$, where
$\Lambda=[0,L]^3$ with periodic boundary conditions; point positions are $\vecX
= (\vecx_1, \ldots, \vecx_N)$ for $\vecx_1, \ldots, \vecx_N \in \Lambda$.  The
Hamiltonian takes one of two forms.  In the first, relevant to the Bose gas, we
have
\begin{align}
\label{eqn:results_Bose_H}
	H_B(\vecX,\pi) &= \frac{T}{4} \sum_{i=1}^N \|\vecx_i-\vecxpii\|^2
		+ \sum_{i<j} V(\vecx_{i}, \vecx_{\pi(i)}, \vecx_{j}, \vecx_{\pi(j)})
\end{align}
where $T=1/\beta$ and the $V$ terms are interactions between permutation jumps.
(The temperature scale factor $T/4$, not $\beta/4$, is surprising but correct
for the Bose-gas derivation of the Hamiltonian.) In the second form of the
Hamiltonian, considered in this paper, we use interactions which are dependent
solely on cycle lengths:
\begin{align}
\label{eqn:results_RCM_H}
	H(\vecX,\pi) &= \frac{T}{4} \sum_{i=1}^N \|\vecx_i-\vecxpii\|^2
		+ \sum_{\ell=1}^N \alpha_\ell r_\ell(\pi),
\end{align}
where $r_\ell(\pi)$ is the number of $\ell$-cycles in $\pi$ and the
$\alpha_\ell$'s are free parameters, called \emph{cycle weights}.  One
ultimately hopes to choose the $\alpha_\ell$'s appropriately for the Bose gas;
even if not, the model is well-defined and of its own interest.

Different choices of $\alpha_\ell$ result in different models: The
\emph{non-interacting model} \cite{bib:gru} has $\alpha_\ell \equiv 0$.  The
\emph{two-cycle model} \cite{bib:qmath,bib:bu07}, has $\alpha_2=\alpha$ and
other cycle weights equal to zero.  The \emph{general-cycle model} has no
restrictions on $\alpha_\ell$.  In \cite{bib:bu08}, the \emph{small
cycle-weight case} is considered:  the only restriction on $\alpha_\ell$ is
that $\alpha_\ell$ goes to zero in $\ell$ faster than $1/\log\ell$.  Lastly,
the \emph{Ewens model}, treated in my research (see also \cite{bib:ewens}) has
$\alpha_\ell \equiv \alpha$ constant in $\ell$.

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics[scale=0.5]{figures/vigre_f_I_alpha_0.0.eps}
\includegraphics[scale=0.5]{figures/vigre_f_S_alpha_0.0.eps}
\caption{Order parameters $f_I$ and $f_S$ for finite systems.
	\label{fig:fI_fS}}
\end{center}
\end{figure}

One may hold point positions fixed, e.g. on the unit lattice; this approach has
been taken for all simulations done up to the present, including specifically
the work described in this paper.  One obtains a Gibbs probability distribution
on $\SN$:
\begin{align}
\label{eqn:results_RCM_P}
	Y(\Lambda,\vecX) &= {\sum_{\sigma\in\SN} e^{-H(\vecX,\sigma)}}, &
	P_{\Lambda,\vecX}(\pi) &= \frac{e^{-H(\vecX,\pi)}} {Y(\Lambda,\vecX)}.
\end{align}
For a random variable $S(\pi)$, we then have
\begin{align}
\label{eqn:E_pi}
	\bbE_{\Lambda,\vecX}[S] = \frac{1}{Y(\Lambda,\vecX)}
		\sum_{\pi\in\SN} S(\pi) e^{-H(\vecX,\pi)},
	\qquad\textrm{i.e.}\qquad
	\bbE_\pi[S(\pi)] = \sum_{\pi\in\SN} P(\pi) S(\pi).
\end{align}
An \emph{order parameter} is the expectation over $\SN$ of a random variable,
dependent on $T$ and $\alpha$, which in the infinite limit is zero to one side
of a \emph{critical temperature} $T_c$ and non-zero on the other side, with
analyticity at all $T \ne T_c$.  See figure \ref{fig:fI_fS} for finite-size
approximations to two such order parameters, $f_I$ and $f_S$: the former is the
fraction of sites in long cycles, whose specific computation is described in
\cite{bib:gru}.  The latter is a scaled winding number, described in
\cite{bib:pc87}.  It quantifies wrapping of long cycles around the 3-torus,
which is the three-dimensional $L$-box with periodic boundary conditions.  Yet
another order parameter is $\bbE[\ellmax]/N$, where $\ellmax(\pi)$ is the
number of sites in the longest cycle of $\pi$.

Shepp and Lloyd \cite{bib:shepplloyd} showed in 1966 that, for uniformly
weighted non-spatial permutations in $\SN$, $\bbE[\ellmax]/N \approx 0.6243$;
unpublished work of Betz and Ueltschi has found $\bbE[\ellmax]/N f_I$ is that
same number for the non-interacting case $\alpha_\ell \equiv 0$.  That is, if
one adds a spatial structure to the permutations, yet restricts consideration
to the number $N f_I$ of points in long cycles, it appears that long cycles are
in some sense (as yet to be clearly defined) uniformly distributed with support
on the points which participate in long cycles.  This issue was also given
preliminary experimental treatment in \cite{bib:gru}.  Thus, we define the
\emph{GRU quotient} \cite{bib:gru} to be the quantity $\bbE[\ellmax]/N f_I$ and
ask whether, for fixed $\alpha$, it is indeed constant in $T$.  (Note that this
quotient is only defined for $T<T_c$; for $T>T_c$, $f_I$ is zero.)

A final order parameter is obtained as follows.  We define a \emph{correlation
length} $\xi$ to be the expected value of the spatial length of cycles.  The
reciprocal correlation length $1/\xi$ is a right-sided order parameter --- a
mirror image of the ones shown in figure \ref{fig:fI_fS} --- permitting
bracketing of the critical temperature from the left and from the right.

%% ----------------------------------------------------------------
\subsection{Current work}

% The worm algorithm's stopping time (described in my summer/fall 2009
% application) is too long for use beyond trivially small system sizes of
% $N=1000$ or so.  Thus, we content ourselves with even winding numbers.  In
% progress is a proof that this reduced sample space does not change the
% critical temperature.

Given the ability to correctly determine the finite order parameter
$S_L(T,\alpha)$, where $S$ is any of several order parameters, one wishes to
take the $L \to \infty$ limit to find $S(T,\alpha)$ and from that find
$T_c(\alpha)$.  I am employing a finite-size scaling technique adapted from
several PIMC studies \cite{bib:nl04,bib:pc87,bib:pgp08,bib:pr92}; see also
\cite{bib:pv} for a nice survey.  Finite-size scaling takes the form of a
hypothesis, or rather a set of hypotheses, which is tested against the data.
Namely, define $t=(T-T_c)/T_c$ and examine, say, $0.99 < t < 1.01$.

The first hypothesis is that, in the infinite-volume limit, the correlation
length $\xi(T,\alpha)$ and the order parameter $S(T,\alpha)$ follow power laws
\begin{align*}
	\xi(T,\alpha) \sim |t|^{-\nu}, \quad T \to T_c
	\qqand
	S(T,\alpha) \sim t^{\rho}, (-t)^{\rho}, \qor |t|^\rho.
\end{align*}
One moreover hypothesizes that for $T$ near $T_c$, $S_L(T)$ and $S(T)$ are
related by a \emph{universal function} $Q$ which depends only on the ratio
$L/\xi$:
\begin{align*}
	S_L(T,\alpha) &= L^{-\rho/\nu} Q(L^{1/\nu} t)
	\sim L^{-\rho/\nu} Q((L/\xi)^{1/\nu}).
\end{align*}

For my dissertation, testing of the finite-size-scaling hypothesis will be done
as follows:

\begin{itemize}

\item First, collect MCMC experimental data with error bars (found using the
method of integrated autocorrelation time \cite{bib:berg}) for a range of
$L$'s, $T$'s, and $\alpha$'s.

\item Second, estimate the critical exponents: given an order-parameter plot
such as \ref{fig:fI_fS}, vary the trial exponent $\hat\rho$.  Raise the raw
data to the $1/\hat\rho$ power.  Find the $\hat\rho$ with least
linear-regression uncertainty.  Do the same for $\hat\nu$.

\item Third, use the \emph{crossing method} \cite{bib:pv} to find $T_c$: once
the exponents are known, plot $L^{\rho/\nu} S_L(T)$ as a function of $T$.
Since at $T=T_c$ we have $t=0$ and $$ L^{\rho/\nu} S_L(T)  = Q(0), $$
regardless of $L$, these curves will cross (approximately, due to sampling
variability) at $T=T_c$.  If they do not, the finite-size-scaling hypothesis is
not verified.

\item Fourth, having estimated $\rho$, $\nu$, and $T_c$, plot $L^{\rho/\nu}
S_L(T)$ as a function of $L^{1/\nu}t$.  This is a plot of the scaling function
$Q$.  If the hypothesis is correct, the curves for all $L$ should coincide, or
collapse, to within sampling error.

\end{itemize}

This determination of critical exponents above will allow me to confirm or
refute, on a confidence-interval basis, the hypothesis that the GRU quotient
$\bbE[\ellmax]/N f_I$ is constant in $T$ for $T<T_c$.  Namely, if the critical
exponents of $\bbE[\ellmax]/N$ and $f_I$ are equal to within sampling error,
one may say that their ratio is constant in $T$.

My research goals for the spring, in addition to other activities as
detailed as the start of this proposal, are quite precise:

\begin{itemize}

\item Complete high-performance computing runs.

\item Use these to update, with smaller error bars, experimental determination
of critical temperature $T_c$ as a function of $\alpha$ which will have been
crudely computed by the end of the fall semester.

\item Use the estimated critical exponents for $\bbE[\ellmax]/N$ and $f_I$ to
test the hypothesis that the GRU quotient $\bbE[\ellmax]/N f_I$ is constant in
$T$ for $T<T_c$.

\item Complete the results section of my dissertation; tie up all loose ends in
my dissertation.

\item Off-lattice computations, as well as use of non-Ewens cycle weights, are
a postdoctoral research topic.

\end{itemize}

%% ----------------------------------------------------------------
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% bib:suto2
% bib:bu
% bib:qmath
% bib:bu07
% bib:bu08
% bib:gru
% bib:ewens
% bib:pc87
% bib:shepplloyd
% bib:nl04
% bib:pgp08
% bib:pr92
% bib:cggp
% bib:berg

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\end{thebibliography}

\end{document}
