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Gibbs Ensembles of Nonintersecting Paths

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Abstract

We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant.

The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices.

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Authors and Affiliations

  1. Department of Mathematics, Caltech, Pasadena, USA

    Alexei Borodin

  2. IITP, RAS, Moscow, Russia

    Alexei Borodin & Senya Shlosman

  3. Centre de Physique Theorique, CNRS, Luminy, Marseille, France

    Senya Shlosman

Authors
  1. Alexei Borodin
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  2. Senya Shlosman
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Correspondence to Alexei Borodin.

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Communicated by H. Spohn

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Borodin, A., Shlosman, S. Gibbs Ensembles of Nonintersecting Paths. Commun. Math. Phys. 293, 145–170 (2010). https://doi.org/10.1007/s00220-009-0906-1

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  • DOI: https://doi.org/10.1007/s00220-009-0906-1

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