We show that with respect to the Haar state, the joint distributions of the generators of Van Daele and Wangs free orthogonal quantum groups are modeled by free families of generalized circular elements and semicircular elements in the large (quantum
) dimension limit. We also show that this class of quantum groups acts naturally as distributional symmetries of almost-periodic free Araki-Woods factors.
We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sa
nov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Steins method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.
To analyze whole-genome genetic data inherited in families, the likelihood is typically obtained from a Hidden Markov Model (HMM) having a state space of 2^n hidden states where n is the number of meioses or edges in the pedigree. There have been sev
eral attempts to speed up this calculation by reducing the state-space of the HMM. One of these methods has been automated in a calculation that is more efficient than the naive HMM calculation; however, that method treats a special case and the efficiency gain is available for only those rare pedigrees containing long chains of single-child lineages. The other existing state-space reduction method treats the general case, but the existing algorithm has super-exponential running time. We present three formulations of the state-space reduction problem, two dealing with groups and one with partitions. One of these problems, the maximum isometry group problem was discussed in detail by Browning and Browning. We show that for pedigrees, all three of these problems have identical solutions. Furthermore, we are able to prove the uniqueness of the solution using the algorithm that we introduce. This algorithm leverages the insight provided by the equivalence between the partition and group formulations of the problem to quickly find the optimal state-space reduction for general pedigrees. We propose a new likelihood calculation which is a two-stage process: find the optimal state-space, then run the HMM forward-backward algorithm on the optimal state-space. In comparison with the one-stage HMM calculation, this new method more quickly calculates the exact pedigree likelihood.
We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one
step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.
