774,185 research outputs found
Hard rods: statistics of parking configurations
We compute the correlation function in the equilibrium version of R\'enyi's
{\sl parking problem}. The correlation length is found to diverge as
when (maximum density) and as
when (minimum density).Comment: 9 pages, 1 figur
A bijection between type D_n^{(1)} crystals and rigged configurations
Hatayama et al. conjectured fermionic formulas associated with tensor
products of U'_q(g)-crystals B^{r,s}. The crystals B^{r,s} correspond to the
Kirillov--Reshetikhin modules which are certain finite dimensional
U'_q(g)-modules. In this paper we present a combinatorial description of the
affine crystals B^{r,1} of type D_n^{(1)}. A statistic preserving bijection
between crystal paths for these crystals and rigged configurations is given,
thereby proving the fermionic formula in this case. This bijection reflects two
different methods to solve lattice models in statistical mechanics: the
corner-transfer-matrix method and the Bethe Ansatz.Comment: 38 pages; version to appear in J. Algebr
Explicit Non-Abelian Monopoles and Instantons in SU(N) Pure Yang-Mills Theory
It is well known that there are no static non-Abelian monopole solutions in
pure Yang-Mills theory on Minkowski space R^{3,1}. We show that such solutions
exist in SU(N) gauge theory on the spaces R^2\times S^2 and R^1\times S^1\times
S^2 with Minkowski signature (-+++). In the temporal gauge they are solutions
of pure Yang-Mills theory on T^1\times S^2, where T^1 is R^1 or S^1. Namely,
imposing SO(3)-invariance and some reality conditions, we consistently reduce
the Yang-Mills model on the above spaces to a non-Abelian analog of the \phi^4
kink model whose static solutions give SU(N) monopole (-antimonopole)
configurations on the space R^{1,1}\times S^2 via the above-mentioned
correspondence. These solutions can also be considered as instanton
configurations of Yang-Mills theory in 2+1 dimensions. The kink model on
R^1\times S^1 admits also periodic sphaleron-type solutions describing chains
of n kink-antikink pairs spaced around the circle S^1 with arbitrary n>0. They
correspond to chains of n static monopole-antimonopole pairs on the space
R^1\times S^1\times S^2 which can also be interpreted as instanton
configurations in 2+1 dimensional pure Yang-Mills theory at finite temperature
(thermal time circle). We also describe similar solutions in Euclidean SU(N)
gauge theory on S^1\times S^3 interpreted as chains of n
instanton-antiinstanton pairs.Comment: 16 pages; v2: subsection on topological charges added, title
expanded, some coefficients corrected, version to appear in PR
Unavoidable Multicoloured Families of Configurations
Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any
there is a constant such that any set system with at least sets
reduces to a -star, an -costar or an -chain. They proved
. Here we improve it to for some constant
.
This is a special case of the following result on the multi-coloured
forbidden configurations at 2 colours. Let be given. Then there exists a
constant so that a matrix with entries drawn from with
at least different columns will have a submatrix that
can have its rows and columns permuted so that in the resulting matrix will be
either or (for some ), where
is the matrix with 's on the diagonal and 's else
where, the matrix with 's below the diagonal and
's elsewhere. We also extend to considering the bound on the number of
distinct columns, given that the number of rows is , when avoiding a matrix obtained by taking any one of the matrices above
and repeating each column times. We use Ramsey Theory.Comment: 16 pages, add two application
A crystal to rigged configuration bijection and the filling map for type
We give a bijection from rigged configurations to a tensor product of
Kirillov--Reshetikhin crystals of the form and in type
. We show that the cocharge statistic is sent to the energy
statistic for tensor products and
. We extend this bijection to a single ,
show that it preserves statistics, and obtain the so-called
Kirillov--Reshetikhin tableaux model for . Additionally, we show that
commutes with the virtualization map and that is naturally a
virtual crystal in type , thus defining an affine crystal structure
on rigged configurations corresponding to .Comment: 40 pages, 6 figures; various revisions from referee comments and
fixed minor typo
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