Free fermion six vertex model: symmetric functions and random domino tilings

• Amol Aggarwal ^[1] ; Alexei Borodin ^[2] ; Leonid Petrov ^[3] ; Michael Wheeler ^[4]
  1. [1] Columbia University

     Columbia University

     Estados Unidos

     
  2. [2] Massachusetts Institute of Technology

     Massachusetts Institute of Technology

     City of Cambridge, Estados Unidos

     
  3. [3] University of Virginia

     University of Virginia

     Estados Unidos

     
  4. [4] University of Melbourne

     University of Melbourne

     Australia

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 3, 2023
• Idioma: inglés
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  • Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions Fλ, Gλ are defined as certain partition functions of the six vertex model, with variables corresponding to rowrapidities, and the labeling signatures λ = (λ1 ≥ ... ≥ λN ) ∈ ZN encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials.

    Cauchy type summation identities for Fλ, Gλ and their skew counterparts follow from the Yang–Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for Fλ and a Sergeev–Pragacz type formula for Gλ. In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions.

    We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard–Mehta type, and the second uses non-standard, inhomogeneous versions of fermionic operators in a Fock space coming from the algebraic Bethe Ansatz for the six vertex model. We also interpret our determinantal processes as random domino tilings of a half-strip with inhomogeneous domino weights. In the bulk, we show that the lattice asymptotic behavior of such domino tilings is described by a new determinantal point process on Z2, which can be viewed as an doubly-inhomogeneous generalization of the extended discrete sine process.