Abstract
We study the central limit theorem for a class of coloured graphs. This means that we investigate the limit behavior of certain random variables whose values are combinatorial parameters associated to these graphs. The techniques used at arriving this result comprise combinatorics, generating functions, and conditional expectations.
Highlights
In this paper we want to verify the central limit theorem CLT in the context of a combinatorial problem for coloured hard dimer configurations, which comprise a certain class of labeled graphs on R
We study the central limit theorem for a class of coloured graphs
This means that we investigate the limit behavior of certain random variables whose values are combinatorial parameters associated to these graphs
Summary
In this paper we want to verify the central limit theorem CLT in the context of a combinatorial problem for coloured hard dimer configurations, which comprise a certain class of labeled graphs on R. We will consider two combinatorial parameters that characterize our hard dimers and will investigate a bivariate mass function. For a given CHDC D, let nb D , nr D denote the numbers of blue, red dimers and nbr D the number of inner vertices, that is, vertices inside dimers that are not boundary points. Due to the symmetry w.r.t. nb and nr, this will lead us to the joint mass function of the r.v.s nb nr, nb nr γb γr , where γb D and γr D denote the number of blue and red vertices which are not occupied by dimers “single points”. The li√mit distribution is a bivariate Gaussian distribution with correlation coefficient equal to −1/ 3
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