Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

Abstract

A Gelfand–Tsetlin scheme of depth \(N\) is a triangular array with \(m\) integers at level \(m\), \(m=1,\ldots ,N\), subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed \(N\)th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its \(q\)-deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

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Notes

  1. 1.

    Here and below \(1_{\{\cdot \cdot \cdot \}}\) denotes the indicator of a set, and \((y)_m:=y(y+1)\ldots (y+m-1)\), \(m=1,2,\ldots \) (with \((y)_0:=1\)) is the Pochhammer symbol.

  2. 2.

    This equation has degree \(k+1\), but it always has the root \(\Omega =1\), so in fact \(\Omega \) satisfies a degree \(k\) equation. There are other (sometimes more suitable) complex parameters of the limit shape and the frozen boundary—\({\mathsf {w}}_c\) (2.10) and \({\mathcal {T}}\) (Remark 2.8). See also Sects. 7.67.7.

  3. 3.

    In the context of random tilings, “turning point” sometimes stays for the point where two types of frozen regions meet (e.g., see [38]); thus, one can think that tangent point is a particular case of a turning point.

  4. 4.

    For example, one can conjugate the kernel as \(K(x_1,n_1;x_2,n_2)\mapsto \frac{f(x_1,n_1)}{f(x_2,n_2)}K(x_1,n_1;x_2,n_2)\) with a nonvanishing function \(f(x,n)\), which does not affect the correlation functions (2.5).

  5. 5.

    Note that \((\chi ,\eta )\in {\mathcal {P}}\) in particular means that \(\chi <b_k\).

  6. 6.

    In fact, the case when there are infinitely many such points (i.e., when a horizontal part of the graph on Fig. 13 is lying at the horizontal coordinate line) corresponds to \((\chi ,\eta )\) belonging to the frozen boundary, when \(S\) has a real double critical point.

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Acknowledgments

The author would like to thank Alexei Borodin for fruitful discussions, and Vadim Gorin for helpful comments. I am also grateful to the anonymous referee for remarks which helped to improve the presentation. The work was partially supported by the RFBR-CNRS grants 10-01-93114 and 11-01-93105.

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Petrov, L. Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes. Probab. Theory Relat. Fields 160, 429–487 (2014). https://doi.org/10.1007/s00440-013-0532-x

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Mathematics Subject Classification

  • 60C05
  • 60G55
  • 82C22