169,241 research outputs found
Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs
Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random
partitions. We connect these random partitions to the Hall-Littlewood
polynomials of symmetric function theory, and use this connection to give
combinatorial proofs of properties of these random partitions. In addition, we
use Markov chains to give an algorithm for generating these partitions.Comment: 10 page
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Geometric generalizations of the Tonnetz and their relation to Fourier phase space
Some recent work on generalized Tonnetze has examined the topologies resulting from Richard Cohn’s common-tone based formulation, while Tymoczko has reformulated the Tonnetz as a network of voice-leading relationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical approach, showing that Tonnetze can always be realized in toroidal spaces,and that the resulting spaces always correspond to one of the possible Fourier phase spaces. We can therefore use the DFT to optimize the given Tonnetz to the space (or vice-versa). I interpret two-dimensional Tonnetze as triangulations of the 2-torus into regions associated with the representatives of a single trichord type. The natural generalization to three dimensions is therefore a triangulation of the 3-torus. This means that a three-dimensional Tonnetze is, in the general case, a network of three tetrachord-types related by shared trichordal subsets. Other Tonnetze that have been proposed with bounded or otherwise non-toroidal topologies, including Tymoczko’s voice-leading Tonnetze, can be under-stood as the embedding of the toroidal Tonnetze in other spaces, or as foldings of toroidal Tonnetze with duplicated interval types.Accepted manuscrip
Stein's Method, Jack Measure, and the Metropolis Algorithm
The one parameter family of Jack(alpha) measures on partitions is an
important discrete analog of Dyson's beta ensembles of random matrix theory.
Except for special values of alpha=1/2,1,2 which have group theoretic
interpretations, the Jack(alpha) measure has been difficult if not intractable
to analyze. This paper proves a central limit theorem (with an error term) for
Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we
recover a known central limit theorem on the distribution of character ratios
of random representations of the symmetric group on transpositions. The case
alpha=2 gives a new central limit theorem for random spherical functions of a
Gelfand pair. The proof uses Stein's method and has interesting ingredients: an
intruiging construction of an exchangeable pair, properties of Jack
polynomials, and work of Hanlon relating Jack polynomials to the Metropolis
algorithm.Comment: very minor revisions; fix a few misprints and update bibliograph
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