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%please enter name of speaker here
\centerline {\textsc {
John Kerl
} }

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%please enter affilitation of speaker here
\centerline{
University of Arizona
}

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%please enter title here
\begin{center} \bf{

Numerical methods for random spatial permutations

} \end{center}

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%please enter abstract here
Abstract:
%Abstract:
%* Cite BU
%* Models of spatial permutations; onset at critical temperature
%  of long cycles in spite of local jumps.
%* Conjectures
%* MCMC/FSS
%* Results
%
Models of random spatial permutations arise in the study of Bose-Einstein
condensation.  Namely, permutations of sites occur with probabilities depending
on lengths of permutation jumps, as well as on interactions between jumps.
Below a critical temperature, one observes the onset of long permutation cycles
in spite of short-distance permutation-jump interactions.  Following work of
Betz, Ueltschi, Gandolfo, and Ruiz \cite{BU2,BU1,U,GRU}, we present conjectures for some of these
models on the cubic unit lattice, along with results obtained by Markov chain
Monte Carlo simulations and finite-size scaling.  These conjectures include the
shift in critical temperature as a function of interaction strength, and
statistics of maximum cycle length in comparison to uniform random
permutations.  We compare our findings to analytical results which are known
for related models.  We include algorithmic approaches to determination of
winding numbers when periodic boundary conditions are placed on the lattice.

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%Topics:
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%-- REMEMBER: NARRATIVE FLOW --
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%* Conjectures
%  - Delta T_c for N_2, vs. known continuum case
%  - Delta T_c for N_ell
%  - ell_max / N_I = ell_max / (N f_I)
%* FSS: theory and results
%  - Delta T_c for N_2, vs. known continuum case
%  - Delta T_c for N_ell
%  - ell_max / N_I = ell_max / (N f_I)


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%Bibliography
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\begin{thebibliography}{}

\bibitem{BU2}
 \textsc{Betz, V.; Ueltschi, D.:} %author(s)
 Spatial random permutations with small cycle weights. %title
 \texttt{arXiv:0812:0569}.

% Spatial random permutations and infinite cycles, with V. Betz, Commun. Math. Phys. 285, 469-501 (2009)

\bibitem{BU1}
 \textsc{Betz, V.; Ueltschi, D.:} %author(s)
 Spatial random permutations and infinite cycles. %title
 \textit{Commun. Math. Phys.}, %name of the journal abbreviated according to AMS
 \textbf{285}, %volume number
 no. 2,      %number
 469--501,      %pages
 (2009).     %year

% The model of interacting spatial permutations and its relation to the Bose
% gas, in Mathematical Results in Quantum Mechanics, pp 255-272, World
% Scientific (2008)

\bibitem{U}
 \textsc{Ueltschi, D.:} %author(s)
 The model of interacting spatial permutations and its relation to the Bose gas. %title
 \textit{Mathematical Results in Quantum Mechanics}, %name of the journal abbreviated according to AMS
 255--272,      %pages
 World Scientific,
 (2008).     %year

% On a model of random cycles, with D. Gandolfo, J. Ruiz, J. Statist. Phys.
% 129, 663-676 (2007)

\bibitem{GRU}
 \textsc{Gandolfo, D.; Ruiz, J.; Ueltschi, D.:} %author(s)
 On a model of random cycles. %title
 \textit{J. Statist. Phys.}, %name of the journal abbreviated according to AMS
 \textbf{129}, %volume number
 no. 4,      %number
 663--676,      %pages
 (2007).     %year

\end{thebibliography}

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