Critical behavior for the model of random spatial permutations
Ph.D. final oral dissertation defense
John Kerl
We elaborate on a model of random spatial permutations, wherein permutations
are weighted according to point positions. This model originates in a study of
the interacting Bose gas; the low-temperature-dependent onset of the appearance
of arbitrarily long cycles is connected to the phase transition of
Bose-Einstein condensates. For our work, we consider a simplified model with
point positions held fixed on the cubic lattice, with interactions expressed as
Ewens-type weights on cycle lengths of permutations. The critical temperature
of the transition to long cycles depends on an interaction-strength parameter
α. For weak interactions, the shift in critical temperature is expected to
be linear in α with constant of linearity c. Using Markov chain
Monte Carlo methods, we find c ≈ 0.618 ± 0.086. This
finding matches a similar analytical result of Ueltschi and Betz. We also
examine the mean longest cycle length as a fraction of the number of sites in
long cycles, recovering and extending an earlier result of Shepp and Lloyd for
non-spatial permutations.
Monday, March 22, 2010, at 10:00 a.m.
Math 402.