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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 spatial permutations arise in the study of Bose-Einstein
condensation.  Below a critical temperature, one observes the onset of long
permutation cycles in spite of short-distance permutation-jump interactions.
Following work of Betz and Ueltschi [cite], 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.

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Topics:

-- REMEMBER: NARRATIVE FLOW --

* It's a math (not physics) conference.
* Models with xrefs
  - non-interacting
  - N_2
  - N_ell; mention cycle weights for BEC not known; intrinsic prb interest
  - V4:  ~PIMC; mess
* Appeals of model:
  - Ease of computation; BBs integrated out.
  - One can *obtain* an RCM via integrating out BBs from BEC.
  - Or, one can *declare* an RCM and examine it for its own intrinsic interest.
* There are only local jumps at all temperatures of interest
  (incl. mean/distr of j0len), yet long cycles occur below T_c.  Incl.
  pictures.  Show some pictures of this.
  - Histograms of jump length for a few T's.
  - Means of jump length
  - Dot plots of onset of long cycles
* 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)
* Winding numbers part 1
* Present GK/GKU Metropolis moves
* Winding numbers part 2
  - Entropy argument about evens
  - Mention the band-update algorithm, in analogy with Swendsen-Wang for Ising;
    low acceptance rate (exp(-L)).
* Detailed balance: implementation & theorem
* 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)
* Worm ...
  - Sketch of challenges with it
  - Crossbow?
  - Call for ideas
* What remains to be done
  - Continuum positions (Poisson point process); mention Lebowitz, Lenci,
    and Spohn.
  - Odd winding numbers, somehow