774,184 research outputs found

    Hard rods: statistics of parking configurations

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    We compute the correlation function in the equilibrium version of R\'enyi's {\sl parking problem}. The correlation length is found to diverge as 21π2(1ρ)22^{-1}\pi^{-2}(1-\rho)^{-2} when ρ1\rho\nearrow1 (maximum density) and as π2(2ρ1)2\pi^{-2}(2\rho-1)^{-2} when ρ1/2\rho\searrow1/2 (minimum density).Comment: 9 pages, 1 figur

    A bijection between type D_n^{(1)} crystals and rigged configurations

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    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

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    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

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    Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any kk there is a constant f(k)f(k) such that any set system with at least f(k)f(k) sets reduces to a kk-star, an kk-costar or an kk-chain. They proved f(k)<(2k)2kf(k)<(2k)^{2^k}. Here we improve it to f(k)<2ck2f(k)<2^{ck^2} for some constant c>0c>0. This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let rr be given. Then there exists a constant crc_r so that a matrix with entries drawn from {0,1,...,r1}\{0,1,...,r-1\} with at least 2crk22^{c_rk^2} different columns will have a k×kk\times k submatrix that can have its rows and columns permuted so that in the resulting matrix will be either Ik(a,b)I_k(a,b) or Tk(a,b)T_k(a,b) (for some ab{0,1,...,r1}a\ne b\in \{0,1,..., r-1\}), where Ik(a,b)I_k(a,b) is the k×kk\times k matrix with aa's on the diagonal and bb's else where, Tk(a,b)T_k(a,b) the k×kk\times k matrix with aa's below the diagonal and bb's elsewhere. We also extend to considering the bound on the number of distinct columns, given that the number of rows is mm, when avoiding a tk×kt k\times k matrix obtained by taking any one of the k×kk \times k matrices above and repeating each column tt times. We use Ramsey Theory.Comment: 16 pages, add two application

    A crystal to rigged configuration bijection and the filling map for type D4(3)D_4^{(3)}

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    We give a bijection Φ\Phi from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form Br,1B^{r,1} and B1,sB^{1,s} in type D4(3)D_4^{(3)}. We show that the cocharge statistic is sent to the energy statistic for tensor products i=1NBri,1\bigotimes_{i=1}^N B^{r_i,1} and i=1NB1,si\bigotimes_{i=1}^N B^{1,s_i}. We extend this bijection to a single Br,sB^{r,s}, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for Br,sB^{r,s}. Additionally, we show that Φ\Phi commutes with the virtualization map and that B1,sB^{1,s} is naturally a virtual crystal in type D4(1)D_4^{(1)}, thus defining an affine crystal structure on rigged configurations corresponding to B1,sB^{1,s}.Comment: 40 pages, 6 figures; various revisions from referee comments and fixed minor typo
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