The shape of a random affine Weyl group element and random core partitions

• Lam, Thomas ^[1]
  1. [1] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 4, 2015, págs. 1643-1662
• Idioma: inglés
• DOI: 10.1214/14-AOP915
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• Resumen

  • Let W be a finite Weyl group and W^ be the corresponding affine Weyl group. We show that a large element in W^, randomly generated by (reduced) multiplication by simple generators, almost surely has one of |W|-specific shapes. Equivalently, a reduced random walk in the regions of the affine Coxeter arrangement asymptotically approaches one of |W|-many directions. The coordinates of this direction, together with the probabilities of each direction can be calculated via a Markov chain on W.

    Our results, applied to type A~n−1, show that a large random n-core obtained from the natural growth process has a limiting shape which is a piecewise-linear graph. In this case, our random process is a periodic analogue of TASEP, and our limiting shapes can be compared with Rost’s theorem on the limiting shape of TASEP.