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\begin{document}
\title{VIGRE Report Part II $\cdot$ Summer 2009}
\author{John Kerl}
%\date{\today}
\date{August 18, 2009}
\maketitle

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\section{Sims Tucson FRG workshop $\cdot$ June}

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.  In this talk, I brought my written presentation up to my current
level of understanding.  The remaining issues involve finite-size scaling; the
only remaining obstacle to publication of this percolation work is my
increasing understanding of finite-size scaling, which is hand-in-hand with FSS
as applied to my dissertation research.

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\section{Marseille $\cdot$ July}

I visited the Centre National pour R\'echerche Scientifique in Marseille
during July 22-24, 2009.  Daniel Ueltschi was present, having arrived a few
weeks earlier; Daniel Gandolfo and Jean Ruiz were my (very gracious) hosts.

\begin{itemize}

\item Daniel Ueltschi presented his upcoming paper with Volker Betz, in which
they obtain a cycle-weight expression for the Bose gas: namely, the
$\alpha_\ell$'s are chosen (using first-order approximations in scattering
length $a$) to match the Bose-gas Hamiltonian.  Daniel Gandolfo may conduct
simulation efforts using these cycle weights, but not immediately: Ueltschi and
Betz are still checking their work.

\item Gandolfo and Ueltschi gave me valuable feedback on my slides for my
upcoming Berlin talk.

\item Gandolfo has a Hoshen-Koppelmann-like idea for simulating the
cycle-weight model, which, if it works out, will be more efficient than my
existing code.  His idea would give $O(N \log N)$ complexity; mine is currently
$O(N^2)$ but with a very small proportionality constant, which (I believe)
suffices.

\item Gandolfo's explication of the Hoshen-Koppelmann algorithm for percolation
enabled me to understand what I hadn't before; I will use this for new
computational work on my percolation project.

\item Gandolfo gave me a better idea for computing distances on the torus.

\item In spring of 2008 I found the true Bose-gas interactions to be
computationally infeasible, even using lookup tables.  Gandolfo points out that
an angle between permutation jumps can be factored out, leading to reduced
lookup-table sizes.  It may be worth re-examining this.

\item Gandolfo points out that the Binder fourth-order cumulant, with which I
had been experimenting for finite-size scaling, is useful for first-order phase
transitions whereas ours is certainly second-order.

\item Ueltschi and I discussed autocorrelation.  This forced me to
theoretically justify my practical understand of autocorrelation; see below.

\item Ueltschi points out a conjecture for the exact infinite-volume behavior
of the $\bbE[f_{1,k}]$ plots.  I will conduct statistical hypothesis tests
against my simulation data for the Ewens model to verify this conjecture.

\item Gandolfo and I will continue to stay in touch, including sharing software
code; he has offered to give me feedback on the dissertation chapters I sent to
him before the meeting.

\end{itemize}

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\section{Berlin $\cdot$ July}

I delivered a contributed talk at the Conference on Stochastic Processes and
Their Applications in July in Berlin.  I had planned to present my worm
algorithm for the random cycle model; on further reflection, realizing that the
audience was more probabilists and fewer experimentalists, I designed a
presentation which would make the random-cycle model palatable to a
probabilist.

I had a productive conversation with Daniel Ueltschi's colleague Stefan
Grosskinsky; with a PhD student contact of Daniel's at Potsdam I had a fruitful
discussion of Ginibre's path-integral methods.  Most importantly, I have an
idea for a postdoc project, on minimal-distance matching of Poisson point
processes.  This would combine theory of Anders Holroyd (communicated to me by
Tom LaGatta) with to-be-developed MCMC methods somewhat related to my current
research.

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\section{Integration workshop $\cdot$ August}

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.

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\section{Bridge group $\cdot$ Early fall}

Tom Kennedy's bridge group has begun meeting again for the fall as of a week
ago; I have now attended two meetings.   Graduate students include myself,
Michael Gilbert, and Shane Passon.  Four potentially interested undergraduates
attended today's meeting; they will be able to participate as undergraduate
research assistants for pay or for academic credit.  We sketched some potential
areas for involvement, and agreed to meet again next week.  At this point, my
hunch is that two the four undergrads will continue with the bridge group; time
will tell.

My other contribution to this early point has been working with Shane on
vertical integration of computational techniques, namely: makefiles, softwaare
profiling (in which subroutines does one's program spend the most time?),
debugging, and hands-off background execution of jobs.

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\section{Dissertation chapters $\cdot$ spring and summer}

Between the submission of my VIGRE application in early April and the start of
the summer, I had written the worm-algorithm and software-architecture chapters
of my dissertation.  This was at Daniel Ueltschi's request, to prepare for my
meeting with Daniel Gandolfo as described above.

The angle-correlation question alluded to in my VIGRE proposal will not be
pursued:  the upper-bound and lower-bound interaction formulas of Betz and
Ueltschi are far too loose to be of use.

For my Berlin presentation I began to quantify the empirical dependence of the
GRU quotient $\ellmax/N f_I$ on the interaction parameter $\alpha$.

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\section{Finite-size scaling $\cdot$ June}

As mentioned above, finite-size scaling is needed for my disseration research
as well as the percolation project.  Let $t=(T-T_c)/T_c$.  Given an
infinite-volume quantity $S(T)$ scaling as $S(T) \sim |t|^\rho$, and a
correlation length $\xi(T)$ scaling as $\xi(T) \sim |t|^{-\nu}$, the FSS
hypothesis is that the finite-volume quantity is $S_L(T) \sim L^{-\rho/\nu}
Q(L^{1/\nu}t)$, where $Q$ is a (model-dependent) universal scaling function.

For percolation, $p_c$ (playing the role of $T_c$) is well-known, as is $\nu$.
Then, using an intersection method to be detailed in my dissertation, it is
relatively straightforward to find $\rho$, which is the item of interest.

For the random-cycle model, $\nu$ and $\rho$ must be determined experimentally,
but are of only secondary interest to $T_c$ which is the prize.  After juggling
half a dozen approaches over the last few months, I currently plan to find
$\nu$ and $\rho$ by extrapolation methods, then use an intersection method to
find $T_c$.

Specifically, my next round of large-scale compute jobs, early this fall
semester, will do the following:

\begin{itemize}
\item Make use of my recently added batched-mean code, for variance reduction.
\item Use the four order parameters $f_I$, $f_S$, $f_W$, and $1/\xi$.
\item Use a few dozen $T$ values for $0.98 \le T/T_c \le 1.02$.
	(This is a tighter region around $T_c$ than in previous experiments.)
\item Use box size $L$ large enough to get stable extrapolation of
	the critical exponents $\rho$ and $\nu$.
\item Organize the CPU time toward getting more reliable (i.e. lower-variance)
	numbers over few parameter points.
\end{itemize}

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\section{Autocorrelations and batched means $\cdot$ July and August}

Introductory statistics courses, at the undergraduate or graduate level, focus
on the handling of IID samples.  Yet, time-series data acquired in Markov chain
Monte Carlo simulations tend to be highly correlated.  Up until this summer, I
had had a vague idea that quantification of autocorrelation was important; late
last spring Tom Kennedy told me about batched means.

This summer, I have finally bitten the bullet and have gotten a handle on both
these topics:  I have worked out detailed calculations for (a) a particular,
exactly solvable Markov chain with known autocorrelation, and (b) any Markov
chain chain where $\Corr(X_i,X_j) = \exp(-|i-j|/\tau)$ for some scaling
constant $\tau$.  The material is ready to be typed up, and will form an
appendix to my dissertation.  Berg's \emph{Markov Chain Monte Carlo Simulations
and Their Statistical Analysis} has proved to be a very useful reference.  As
well as working out the theory, I have implemented batched means in my software
simulations.  The theoretical understanding allows me to choose the correct
batch sizes.

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\section{Caching of cycle lengths}

A significant software optimization was a change to cache cycle lengths
in a data structure separate from the lattice.  The mathematical results
are the same; however, the program now runs to completion in reasonable
time for larger lattice sizes with Ewens interactions, which was not
the case before.

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\section{Upcoming conferences $\cdot$ early 2010}

The Joint Meetings of the AMS/MAA will be Jan. 13-16, 2010.  I have submitted
my abstract for a contributed talk on my research.  I will be on the job
market; AMS attendance will be key.

David Landau's Center for Simulational Physics in Athens, Georgia, will be
holding a weeklong conference Mon.-Fri. Feb. 15-19 2010.  Applications are not
yet open; I will certainly attend.  This conference looks to be the most
closely aligned with my interests that I have yet attended.  I hope to obtain
feedback toward overcoming the stopping-time limitation of my nascent worm
algorithm; I also hope to make many professional contacts.

\end{document}
