38 research outputs found
On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices
We establish formulae for the moments of the moments of the characteristic
polynomials of random orthogonal and symplectic matrices in terms of certain
lattice point count problems. This allows us to establish asymptotic formulae
when the matrix-size tends to infinity in terms of the volumes of certain
regions involving continuous Gelfand-Tsetlin patterns with constraints. The
results we find differ from those in the unitary case considered previouslyComment: 31 page
Approximate Counting, the Lovasz Local Lemma and Inference in Graphical Models
In this paper we introduce a new approach for approximately counting in
bounded degree systems with higher-order constraints. Our main result is an
algorithm to approximately count the number of solutions to a CNF formula
when the width is logarithmic in the maximum degree. This closes an
exponential gap between the known upper and lower bounds.
Moreover our algorithm extends straightforwardly to approximate sampling,
which shows that under Lov\'asz Local Lemma-like conditions it is not only
possible to find a satisfying assignment, it is also possible to generate one
approximately uniformly at random from the set of all satisfying assignments.
Our approach is a significant departure from earlier techniques in approximate
counting, and is based on a framework to bootstrap an oracle for computing
marginal probabilities on individual variables. Finally, we give an application
of our results to show that it is algorithmically possible to sample from the
posterior distribution in an interesting class of graphical models.Comment: 25 pages, 2 figure