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\begin{document}
\title{VIGRE Application Part II $\cdot$ Summer 2009}
\author{John Kerl}
\date{\today}
\maketitle

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

I am finishing my fourth year of a five-year PhD program.  The period of
support (summer 2009) will lie between my fourth and fifth years.

The spring of 2009 saw the following:

\begin{itemize}

\item I successfully completed my comprehensive examination in January 2009.
I exposited a recent paper of Daniel Ueltschi \cite{bib:qmath}, which my
dissertation research is extending.

\item Ueltschi was present in Tucson on a sabbatical semester; he and I are
collaborating closely on my dissertation research.  In particular, my funding
this semester is from Ueltschi's NSF grant.  (In spring of 2008 I was also
funded under Ueltschi's grant.  The work I did at that time preceded my
comprehensive examination, but that work has led seamlessly into my
dissertation.)

\item I obtained access to the ICE cluster in the University of Arizona's High
Performance Computing center.  This reduces compute time, relative to my
previous use of the math department's chivo cluster, by an order of magnitude.

\item The local March 2009 workshop on Entropy and the Quantum, organized by
Daniel Ueltschi and Robert Sims, drew several visitors from the international
mathematical-physics community.  Among these was Volker Betz, Ueltschi's
co-worker at Warwick and chief collaborator.  During the workshop (as I write
this proposal), Betz and I are exploring a question in bulk flow of permutation
cycles.

\item Also during the workshop, John LaPeyre and I are sketching out a joint
paper which follows on work of Perseguers, Wehr, et al. \cite{bib:entdist}.  My
portion is an independent study project under Jan Wehr in spring of 2008.  I
obtained interesting results that spring, and presented them to the UA
mathematical physics seminar in October 2008; it would benefit me
professionally to write these results up nicely and submit them for
publication.

\item In the spring of 2009, I am taking Bill Faris' course on Discrete-Time
Stochastic Processes, which reinforces my current knowledge of Markov chains.
I am also taking Theory of Statistics from Walter Piegorsch.  This is a core
course in the statistics program; it requires no small amount of effort, but it
completes coursework for my PhD minor in statistics.

\item For fall 2008 and spring 2009, I am organizing the weekly Graduate
Student Colloquium.  I am also serving as this year's student representative to
the department's graduate committee.  Although the committee work per se is
light, I am organizing the weekly grad tea as well as weekly lunches with the
departmental colloquium speaker.  I also contributed significantly to the
department's recruitment workshop in March 2009.

\end{itemize}

The period of support, summer 2009, will include the following:

\begin{itemize}

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

\item I will deliver a contributed talk (already accepted) at the Conference on
Stochastic Processes and Their Applications in July in Berlin.  I will present
my worm algorithm for the random cycle model.

\item As time permits, I may visit Daniel Ueltschi in Marseille before that
conference.

\item Working with John LaPeyre, I will write up the above-described
percolation results for publication.

\item I will assist with the department's integration workshop for incoming
graduate students.

\end{itemize}

My final year at the UA, fall 2009 and spring 2010, includes the following:

\begin{itemize}

\item I will continue my dissertation work, with timely graduation in May 2010.

\item I will present the progress of that work to the UA mathematical physics
seminar in the fall.

\item I will complete my final course --- Probability and Random Processes in
Engineering, in the Electrical Engineering department --- in fall 2009,
satisfying an out-of-department requirement.

\item I plan to teach both fall 2009 and spring 2010.

\item I will, of course, be spending much of the year applying for employment.
Thus, it is my desire to accomplish as much during the summer as possible.
VIGRE support for summer 2009 will help me to achieve that goal.

\end{itemize}

%% ----------------------------------------------------------------
\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 is spending a sabbatical semester at the UA for spring
2009; by the time of his return to Warwick, I will be working largely
independently.  We will continue to communicate; as well, my local advisor, Tom
Kennedy, is up to date on my research and I will be working with him to bring
the dissertation to completion.

%% ----------------------------------------------------------------
\subsection{Historical context}

Bose and Einstein \cite{bib:bose,bib:einstein} predicted in the 1920s that
non-interacting particles with integer spin may collapse into a macroscopic
occupation of the ground state of the external potential.  Einstein predicted a
critical temperature for the phenomenon.  It is a long-standing question to
understand the effects of interbosonic interaction strength $a$ on the
condensate's critical temperature $T_c^{(a)}$.  Several research groups have
investigated BEC theoretically over the last three decades, with widely varying
results on the shift in critical temperature; see \cite{bib:baymetal} for a
survey.  Current consensus within the theoretical physics community
\cite{bib:baymetal} is that
\begin{align}
	\Delta T_c &= \frac{T_c^{(a)}-T_c^{(0)}}{T_c^{(0)}}
\end{align}
is linear in $a$ for small $a$.  The main goal of this project is to determine
the constant of linearity with a high degree of certainty, for various
interaction models.

%One expects the critical combination of parameters to be a manifold in
%$(\rho,\beta,a)$ space, where $\beta=1/T$, so one may fix the density
%$\rho_c^{(a)}$ and obtain a critical temperature $T_c^{(a)}$ or vice versa.

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

One begins with a Hamiltonian for particles with two-body interactions.
Specifically, consider a system of $N$ bosons in a cube $\Lambda \in \R^d$ of
size $L$ and volume $L^d$.  The positions of the particles are either on a unit
lattice, so $N=L^d$ with density $\rho=1$, or continuously distributed
according to a point process with variable density $\rho=N/V$.  The
interparticle potential $U$ is in terms of a scattering length $a$, where $a$
is nominally the radius of a hard-core potential.  The Hamiltonian for
pair-interacting particles is
\begin{align}
\label{eqn:bosonic_H}
	H &= -\sum_{i=1}^N \Delta_i + \sum_{i<j} U(\vecx_i-\vecx_j)
\end{align}
where $\Delta_i$ is the Laplacian and $U$ is a multiplication operator.  This
operator acts in the space $L^2(\Lambda^N)_{\textrm{sym}}$ of symmetric,
square-summable wavefunctions with periodic boundary conditions.

One may write down the partition function $Z_{\Lambda,N} = \Tr(e^{-\beta H})$
and apply a multi-particle Feynman-Kac formula, involving permutation symmetry
of bosonic wave functions, to obtain a Hamiltonian in which permutation jumps
rather than particles interact.  (The derivation is sketched in
\cite{bib:qmath,bib:bu}, and was worked out in full detail in my written
comprehensive paper.) A cluster expansion, to first order in the scattering
length of the particles, yields a Hamiltonian with only jump-pair interactions:
\begin{equation}
\label{eqn:perm_H}
	H_P = \frac{1}{4\beta} \sum_{i=1}^N \|\vecx_i - \vecx_{\pi(i)}\|^2
		+ \sum_{i<j} V(\vecx_i, \vecx_{\pi(i)}, \vecx_j, \vecx_{\pi(j)}).
\end{equation}
The jump-pair interaction is
\begin{equation}
\label{eqn:V}
	V(\vecx_i,\vecx_{\pi(i)},\vecx_j,\vecx_{\pi(j)}) = \int
	\left[ 1 - \e^{-\frac{1}{4\beta}
		\int_0^{4\beta} U(\vecw(s)) \,ds}
	\right]
	\dd
	\vecWhat^{4\beta}_{\vecx_i-\vecx_j, \vecx_{\pi(i)}-\vecx_{\pi(j)}}(\vecw)
\end{equation}
where
$d\vecWhat$ is a unit-normalized Wiener measure on Brownian bridges $\vecw$
running from $\vecx_i-\vecx_j$ to $\vecx_{\pi(i)}-\vecx_{\pi(j)}$ in imaginary
time $4\beta$.  The jump-pair interaction $V(\vecx_i, \vecx_{\pi(i)}, \vecx_j,
\vecx_{\pi(j)})$ may be interpreted as the probability that a Brownian bridge
from $\vecx_i-\vecx_j$ to $\vecx_{\pi(i)}-\vecx_{\pi(j)}$ intersects the ball of
radius $a$.  (See figure \ref{fig:2w_to_1w}.)

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics{figures/2w_to_1w.eps}
\caption{Replacement of double Brownian bridge by single Brownian bridge
	for jump-pair interactions.
	\label{fig:2w_to_1w}}
\end{center}
\end{figure}

Equation \ref{eqn:perm_H} gives rise to a Gibbs distribution $P(\pi) =
e^{-H(\pi)}/Z$ on the $N!$ permutations of the $N$ bosons.  Then, one may
compute the expectation of various random variables in the finite system.  For
\cite{bib:qmath,bib:bu,bib:gru} as well as my computational work in spring
2008, the random variable of interest was the fraction $f_I$ of sites in
macroscopic cycles, namely,
\begin{align}
\label{eqn:f_I}
	f_I(N) = 1 - \sum_{k=1}^{\veps(N)} P(\ell_0=k)
\end{align}
where $\ell_0$ is the number of sites in the permutation cycle containing the
origin and $\veps(N)$ is a small fraction of $N$, say, $\sqrt{N}$.  (This is
chosen so that as $N\to\infty$, one has the fraction of sites in infinite
cycles.) As suggested by figure \ref{fig:fI_fS}, one may use $f_I$ as an order
parameter namely, a quantity which is zero on one side of a phase transition
and non-zero on the other.  For finite systems, the $f_I(L,T)$ is smooth in
$T$; as $L$ increases, the graph approaches a non-analyticity at $T_c$.

\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}
%
%\includegraphics[scale=0.5]{figures/vigre_f_I_alpha_1.0.eps}
%\includegraphics[scale=0.5]{figures/vigre_f_S_alpha_1.0.eps}
\caption{Order parameters $f_I$ and $f_S$ for finite systems.
	\label{fig:fI_fS}}
\end{center}
\end{figure}

Direct use of equation \ref{eqn:V} is prohibitive both analytically and
numerically.  The following approaches have been taken:

\begin{itemize}

\item One may drop interactions altogether.  This model was examined using
Markov chain Monte Carlo methods in \cite{bib:gru}, with particles distributed
on a cubic unit lattice.  One recovers the critical density for the Bose gas in
the non-permutation Hamiltonian (equation \ref{eqn:bosonic_H}), and finds
experimentally that the occurrence of infinite macroscopic cycles corresponds
to condensation as predicted by Feynman \cite{bib:feynman}.  An analogous
result is proved in \cite{bib:bu} for non-interacting particles distributed on
a continuum.

\item One may drop all jump-pair interactions except the most strongly
interacting ones, namely, two-cycles.  Letting $N_2(\pi)$ denote the number of
two-cycles in the permutation $\pi$, one would like to have a simple
Hamiltonian of the form
\begin{align}
	\Halpha_P(\vecX,\pi) &= \sum_{i=1}^N
		\frac{1}{4\beta}\|\vecx_i-\vecxpii\|^2
		+ \alpha N_2(\pi).
\end{align}
Betz and Ueltschi \cite{bib:bu} connect this to equations \ref{eqn:perm_H} and
\ref{eqn:V} via the calculation $\alpha = \left(\frac{8}{\pi\beta}\right)^{1/2}
a + O(a^2)$.  It is also found analytically that
$
	\Delta T_c \approx 0.37 \rho^{1/3}a
$
for particles distributed on the continuum.

\item In \cite{bib:bu08}, the $N_2$ concept is extended to the form
\begin{align}
	\HP(\vecX,\pi)
	&= \sum_{i=1}^N \frac{1}{4\beta}\|\vecx_i-\vecxpii\|^2
	+ \sum_{\ell=2}^N \alpha_\ell N_\ell(\pi)
\end{align}
where $N_\ell(\pi)$ counts the number of $\ell$-cycles in the permutation
$\pi$.  Here, it is not known how to compute the $\alpha_\ell$'s to match the
interactions in equation \ref{eqn:V}.  Nonetheless, one may examine this model
for its intrinsic interest --- say, with $\alpha_\ell$ constant in $\ell$.

\item My attempts to attack equation \ref{eqn:V} using Brownian-bridge
computations in spring of 2008 showed that the approach is computationally
intractable.

\item Betz and Ueltschi have a Riemann-integral approximation to equation
\ref{eqn:V} which I also examined in spring of 2008.  This was better than the
Brownian-bridge approach, but its computational expense was also prohibitive.

\item Recent, unpublished collaboration between Betz and Ueltschi obtains lower
and upper bounds on equation \ref{eqn:V}; these can be computed easily.
Namely, replace the hard-core potential of radius $a$ with the soft-core
potential of height $u$ and radius $b$.  Take $\gamma = ub^3/3$ to be small.
For $V(\vecx_i,\vecx_{\pi(i)},\vecx_j,\vecx_{\pi(j)})$, let $\vecz =
\vecx_j-\vecx_i \qand \vecw = \vecx_{\pi(j)}-\vecx_{\pi(i)}$.  Then
\begin{equation}
\begin{aligned}
	A &=
	\frac{\gamma (\|\vecw\| + \|\vecz\|)}{\|\vecw\|\,\|\vecz\|}
	\exp\left(\frac{-\|\vecz+\vecw\|^2}{8 \beta}\right) \\
%
	B &=
	\frac{\gamma}{\|\vecw\|\vecz\|}
	\exp\left(\frac{-\vecz\cdot\vecw}{4 \beta}\right)
	\left(\|\vecw\| \exp\left(-\frac{\|\vecz\|^2}{4\beta}\right) +
	\|\vecz\| \exp\left(\frac{-\|\vecw\|^2}{4\beta}\right)\right) \\
%
	A &\le
	V(\vecx_i,\vecx_{\pi(i)},\vecx_j,\vecx_{\pi(j)}) = V(\vecz,\vecw)
	\le B.
\end{aligned}
\end{equation}
As suggested by figure \ref{fig:2w_to_1w}, collision probability for Brownian
bridges is increased when paths cross.  Thus, Betz expects permutation jumps to
be preferentially aligned or anti-aligned.  This effect should appear in the
expectation of the cosine of the angle between permutation jumps at neighboring
sites.

\end{itemize}

%%% ----------------------------------------------------------------
%\subsection{Completed work}
%
%In spring of 2008, worked as a research assistant under Daniel Ueltschi, I
%first reproduced the MCMC results of \cite{bib:gru}.  This included
%construction of software architecture which I have been using and extending
%since that time.  (I chose not to adapt the software of Gandolfo and Ruiz, in
%the realized hope that that they and I would obtain similar results using
%independent techniques.) Second, I simulated the $N_2$ model and found that,
%for the $N_2$ model on the lattice, $\Delta T_c \approx 0.6 \rho^{1/3}a$.  This
%contrasts with the analytical result for the $N_2$ model on the continuum.  As
%mentioned above, I attempted to attack equation \ref{eqn:V} using
%Brownian-bridge and Riemann-integral approaches.  In retrospect, I was
%essentially beginning to use a path-integral Monte Carlo approach, as described
%in the next section.  In April 2008, I presented my ongoing work to the UA
%Mathematical Physics seminar.  In summer and fall 2008, I prepared for my
%comprehensive examination.  The main body of this work was to work out, in
%complete detail, the sketch in \cite{bib:qmath} of the derivation of equations
%\ref{eqn:perm_H} and \ref{eqn:V}.

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

I am presently pursuing two threads:  (1) re-working individual MCMC
simulations, each of which produces an order-parameter data point, and (2)
extrapolation and finite-size scaling which permit determination of the
critical temperature from order-parameter data.

In spring 2009, Ueltschi and I began to consider the superfluid fraction $f_S$
rather than $f_I$ (equation \ref{eqn:f_I}) as an order parameter.  As figure
\ref{fig:fI_fS} suggests, the two order parameters have the same critical
temperature but different critical exponents.  The choice of $f_S$ as order
parameter was motivated by consideration of path-integral Monte Carlo (PIMC)
studies \cite{bib:ceperley, bib:kps01, bib:nl04, bib:bps06, bib:pst98,
bib:gcl97, bib:hk99, bib:kps01, bib:pc87, bib:pgp08, bib:pr92}.  (One may think
of PIMC as sampling from Brownian bridges as in equation \ref{eqn:V}.)
Specifically, one has
\begin{align}
\label{eqn:f_S}
	f_S = \langle \vecW^2 \rangle L^2 / 3 \beta N
\end{align}
where the random variable $\vecW = (W_x, W_y, W_z)$ is a triple defined by
\begin{align}
\label{eqn:Wno}
	\vecW(\pi) = \frac{1}{L} \sum_{i=1}^N (\vecx_{\pi(i)} - \vecx_i).
\end{align}
That is, for a given permutation, one simply counts the number of windings
around the 3-torus (with topology emplaced by periodic boundary conditions) in
each direction.

We soon discovered that the Metropolis sampling algorithm of \cite{bib:gru}
suffers from a serious defect:  It creates winding cycles only with opposite
signs, so that permutations are created with zero winding number.  Another
concern (as yet unexplained) is that the superfluid fractions we are seeing
experimentally on the lattice exceed the unit interval.

To sample permutations with non-zero winding number, I reviewed the above-cited
PIMC studies, where the same problem arises, then adapted the PIMC concept of a
\emph{worm algorithm} to our random-cycle model.  Namely, one opens,
rearranges, and closes permutation cycles.  The open cycles are free to wrap
around the torus before closing.  As shown in figure \ref{fig:worm1}, an open
cycle on $N$ points may be viewed as a cycle on $N+1$ points.  I take the
$(N+1)$st point to be \emph{non-spatial}: it lies at no distance from any
lattice point.  This degree of freedom allows Metropolis udpates to explore the
full configuration space of lattice permutations on the 3-torus.

\begin{figure}[!htb]
\begin{center}
\psfragscanon
\includegraphics[scale=0.7]{figures/worm1.eps}
\caption{Open cycles as permutations on $N+1$ points.
	\label{fig:worm1}}
\end{center}
\end{figure}

Precisely, we have
\begin{align}
\label{eqn:worm_H}
	H_P'(\pi) = \sum_{\stackrel{1 \le i \le N}{\pi(i) \ne N+1}}
		\|\vecx_i - \vecx_{\pi(i)}\|^2
	+ \sum_{\stackrel{1 \le i < j \le N}{\pi(i) \ne N+1,\pi(j) \ne N+1}}
		V(\vecx_i,\vecx_{\pi(i)},\vecx_j, \vecx_{\pi(j)}).
\end{align}
That is, the $(N+1)$st point has no distance or interactions associated with
it.  A simple calculation shows that the marginal distribution $P'(\pi \mid
\pi(N+1)=N+1)$ for Gibbs measure on $N+1$ points induced by equation
\ref{eqn:worm_H} is equal to the probability distribution for Gibbs measure on
$N$ points induced by equation \ref{eqn:perm_H}.  As well, I have proved the
detailed-balance condition for Metropolis updates on the extended $(N+1)$-point
system.  Full details will be provided in my dissertation.

(All computations this summer will be done on the lattice.  Point-process
simulations may be performed in the fall, if we deem that to be of interest for
the dissertation.  A Poisson point process is a tractable approach; the true
point process for the Bose gas is very difficult to obtain.)

The second thread of my current efforts, given the ability to correctly
determine the finite order parameter $f_S(T,L)$, is to take the $L \to \infty$
limit.  One technique, which I am developing ad hoc, is to extrapolate
pointwise in $T$ from data such as those in figure \ref{fig:fI_fS} to the true
$f_S$ curve for $L=\infty$.  At this point, it becomes easy, using log-log
plots and linear regression, to read off the critical temperature with error
bars.

In parallel, I am employing a finite-size scaling technique adapted from
several PIMC studies \cite{bib:nl04,bib:pc87,bib:pgp08,bib:pr92}.  Here one
assumes that the finite system's $f_S(T,L)$ is of the form
\begin{align}
	f_S(T,L) &=
		\frac{1}{L}\;
		Q\left( \left(\frac{T-T_c}{T_c}\right) L^{1/\nu}, \zeta L^{-\delta/\nu}
		\right)
\end{align}
for some analytic function $Q$ and some parameters $\nu$, $\zeta$, and
$\delta$.  Taylor-expanding to first order in the small arguments, followed by
some algebra, permits a two-step regression analysis of the MCMC data resulting
in a determination of $T_c$ with error bars.

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

\begin{itemize}

\item Write the worm-algorithm, superfluid-fraction, extrapolation, and
finite-size-scaling chapters of my dissertation.

\item Quantify $\Delta T_c$ for the $N_2$ model on the lattice; compare to the
continuum result in \cite{bib:qmath}.

\item Quantify $\Delta T_c$ for the $N_\ell$ model on the lattice with constant
$N_\ell$.  This topic is completely open in the literature.

\item Investigate with Betz the angle correlation for $V(\vecz,\vecw)$
interactions.  This topic is also completely open in the literature.

\end{itemize}

%% ----------------------------------------------------------------
\begin{thebibliography}{}

% bib:qmath
% bib:entdist
% bib:bose
% bib:einstein
% bib:baymetal
% bib:bu
% bib:gru
% bib:feynman
% bib:bu08
% bib:ceperley
% bib:kps01
% bib:nl04
% bib:bps06
% bib:pst98
% bib:gcl97
% bib:hk99
% bib:pc87
% bib:pgp08
% bib:pr92

\bibitem{bib:qmath}
Ueltschi,~D.
\newblock {\em The model of interaction spatial permutations and its relation
to the Bose gas.}
\newblock \texttt{arXiv:0712.2443v3}.

\bibitem{bib:entdist}
Perseguers, Cirac, Ac\'{i}n, Lewenstein, and Wehr.
\newblock \emph{Entanglement Distribution in Pure-State Quantum Networks.}
\newblock Phys. Rev. A, vol. 77, issue 2, 2008.
\newblock \texttt{arXiv:0708.1025v2}.

\bibitem{bib:bose}
Bose,~S.N.
\newblock{Z. Phys. 26, 178 (1924)}.

\bibitem{bib:einstein}
Einstein,~A.
\newblock{Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 1, 3 (1925)}.

\bibitem{bib:baymetal}
Baym,~G., Blaizot,~J.-P., Holzmann,~M., Lalo\"e,~F., and Vautherin,~D.
\newblock {\em Bose-Einstein transition in a dilute interacting gas}.
\newblock \texttt{arXiv:cond-mat/0107129v2}.

\bibitem{bib:bu}
Betz,~V. and Ueltschi,~D.
\newblock{\em Spatial random permutations and infinite cycles}.
\newblock \texttt{arXiv:0711.1188}.

\bibitem{bib:gru}
Gandolfo,~D., Ruiz,~J., and Ueltschi,~D.
\newblock {\em On a model of random cycles}.
\newblock \texttt{arXiv:cond-mat/0703315}.
\newblock Statist. Phys. 129, 663-676 (2007).

\bibitem{bib:feynman}
Feynman,~R.P.
\newblock {\em Atomic theory of the $\lambda$ transition in Helium}.
\newblock Phys. Rev. 91, 1291–1301, (1953).

\bibitem{bib:bu08}
Betz,~V. and Ueltschi,~D.
\newblock {\em Spatial random permutations with small cycle weights.}
\newblock \texttt{arXiv:0812.0569v1}.

\bibitem{bib:ceperley}
Ceperley,~D.M.
\newblock {\em Path integrals in the theory of condensed helium}.
\newblock Rev. Mod. Phys. vol. 67, no. 2, April 1995.

\bibitem{bib:kps01}
Kashurnikov,~V.A., Prokof'ev,~N.V., and Svistunov,~B.V.
\newblock {\em Critical temperature shift in weakly interacting Bose gas}.
\newblock Phys. Rev. Lett. vol. 87, no. 12, Aug. 2001.

\bibitem{bib:nl04}
Nho,~K. and Landau,~D.P.
\newblock {\em Bose-Einstein condensation temperature of a homogeneous weakly
	interacting Bose gas:  PIMC study}.
\newblock \texttt{arXiv:cond-mat.stat-mech/0404238v1}.

\bibitem{bib:bps06}
Boninsegni,~M., Prokof'ev,~N.V., and Svistunov,~B.V.
\newblock {\em Worm algorithm and diagrammatic Monte Carlo:  A new approach
	to continuous-space path integral Monte Carlo simulations.}
\newblock Physical Review E \textbf{74}, 036701 (2006).

\bibitem{bib:pst98}
Prokof'ev,~N.V., Svistunov,~B.V., and Tupitsyn,~I.S.
\newblock {\em Exact, complete, and universal continuous-time worldline Monte
	Carlo approach to the statistics of discrete quantum systems.}
\newblock Journal of Experimental and Theoretical Physics, vol. 87, no. 2
	(1998).

\bibitem{bib:gcl97}
Gr\"uter,~P. and Ceperley,~D.
\newblock {\em Critical temperature of Bose-Einstein condensation of
	hard-sphere gases.}
\newblock Physical Review Letters, vol. 79, no. 19 (1997).

\bibitem{bib:hk99}
Holzmann,M. and Krauth,~W.
\newblock {\em Transition temperature of the homogeneous, weakly interacting
	Bose gas.}
\newblock Physical Review Letters, vol. 83, no. 14 (1999).

\bibitem{bib:pc87}
Pollock,~E.L. and Ceperley,~D.M.
\newblock {\em Path-integral computation of superfluid densities.}
\newblock Physical Review B, vol. 36, no. 16 (1987).

\bibitem{bib:pgp08}
Pilati,~S., Giorgini,~S., and Prokof'ev,~N.V.
\newblock {\em Critical temperature of interacting Bose gases in two and
	three dimensions.}
\newblock Physical Review Letters \textbf{100}, 140405 (2008).

\bibitem{bib:pr92}
Pollock,~E.L. and Runge,~K.J.
\newblock {\em Finite-size-scaling analysis of a simulation of the
	${}^4$He superfluid transition}.
\newblock Physical Review B, vol. 46, no. 6 (1992).


%\bibitem{bib:boulder}
%Anderson,~M.H., Ensher,~J.R., Matthews,~M.R., Wieman,~C.E., and Cornell,~E.A.
%\newblock {\em Observation of Bose-Einstein Condensation in a Dilute Atomic
%	Vapor}.
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\end{thebibliography}

\end{document}

%%% ================================================================
%xxx lattice vs. continuum?
%
%xxx quenched vs. annealed?
%
%xxx ICE cluster
