A multi-dimensional Markov chain and the Meixner ensemble

Abstract

We show that the transition probability of the Markoc chain $(G(j,1),...,G(j,n){)}_{j\ge 1}$, where the $G(i,j{)}^{\prime }s$ are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for $G(m,n)$. We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

14 pages

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• Dimers and orthogonal polynomials: connections with random matrices
• Skew Howe duality and limit shapes of Young diagrams
• Quantum Mechanics of Plancherel Growth
• Free fermionic probability theory and K-theoretic Schubert calculus
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