933,265 research outputs found
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Counting defects with the two-point correlator
We study how topological defects manifest themselves in the equal-time
two-point field correlator. We consider a scalar field with Z_2 symmetry in 1,
2 and 3 spatial dimensions, allowing for kinks, domain lines and domain walls,
respectively. Using numerical lattice simulations, we find that in any number
of dimensions, the correlator in momentum space is to a very good approximation
the product of two factors, one describing the spatial distribution of the
defects and the other describing the defect shape. When the defects are
produced by the Kibble mechanism, the former has a universal form as a function
of k/n, which we determine numerically. This signature makes it possible to
determine the kink density from the field correlator without having to resort
to the Gaussian approximation. This is essential when studying field dynamics
with methods relying only on correlators (Schwinger-Dyson, 2PI).Comment: 11 pages, 7 figures
- …