713 research outputs found
Improving lattice perturbation theory
Lepage and Mackenzie have shown that tadpole renormalization and systematic
improvement of lattice perturbation theory can lead to much improved numerical
results in lattice gauge theory. It is shown that lattice perturbation theory
using the Cayley parametrization of unitary matrices gives a simple analytical
approach to tadpole renormalization, and that the Cayley parametrization gives
lattice gauge potentials gauge transformations close to the continuum form. For
example, at the lowest order in perturbation theory, for SU(3) lattice gauge
theory, at the `tadpole renormalized' coupling to be compared to the non-perturbative numerical value Comment: Plain TeX, 8 page
Finite Simple Groups as Expanders
We prove that there exist and such that every
non-abelian finite simple group , which is not a Suzuki group, has a set of
generators for which the Cayley graph \Cay(G; S) is an
-expander.Comment: 10 page
Paths in quantum Cayley trees and L^2-cohomology
We study existence, uniqueness and triviality of path cocycles in the quantum
Cayley graph of universal discrete quantum groups. In the orthogonal case we
find that the unique path cocycle is trivial, in contrast with the case of free
groups where it is proper. In the unitary case it is neither bounded nor
proper. From this geometrical result we deduce the vanishing of the first
L^2-Betti number of A_o(I_n).Comment: 30 pages ; v2: major update with many improvements and new results
about the unitary case ; v3: accepted versio
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
Small spectral radius and percolation constants on non-amenable Cayley graphs
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we
study the following question. For a given finitely generated non-amenable group
, does there exist a generating set such that the Cayley graph
, without loops and multiple edges, has non-unique percolation,
i.e., ? We show that this is true if
contains an infinite normal subgroup such that is non-amenable.
Moreover for any finitely generated group containing there exists
a generating set of such that . In particular
this applies to free Burnside groups with . We
also explore how various non-amenability numerics, such as the isoperimetric
constant and the spectral radius, behave on various growing generating sets in
the group
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