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Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

Svetlana Gavrilova, Leonid Petrov

Abstract

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (arXiv:math/9907127 [math.RT]). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are \emph{not} determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the $4\times4$ problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size $n\ge4$ which appear new for $n\ge 5$. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (arXiv:math/9804027 [math.CA]), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

Abstract

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (arXiv:math/9907127 [math.RT]). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are \emph{not} determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size which appear new for . By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (arXiv:math/9804027 [math.CA]), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.
Paper Structure (38 sections, 18 theorems, 120 equations, 13 figures)

This paper contains 38 sections, 18 theorems, 120 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Random partitions from symmetric functions
  3. Grothendieck measures on partitions
  4. Absence of determinantal structure
  5. Tilted biorthogonal ensembles
  6. Limit shape
  7. Outline
  8. Acknowledgments
  9. Tilted biorthogonal ensembles
  10. Definition of the ensemble
  11. Normalization
  12. Two-dimensional process
  13. Determinantal kernel
  14. Marginals and correlations of the tilted biorthogonal ensemble
  15. Grothendieck random partitions
  16. ...and 23 more sections

Key Result

Theorem 1.2

For certain fixed $N$ and values of parameters $x_i,y_j$, and $\beta$, the correlations eq:Schur_corr_f of the Grothendieck measures do not possess a determinantal form. That is, there does not exist a function $K\colon \mathbb{Z}_{\ge0}^{2}\to \mathbb{C}$ for which $\rho(a_1,\ldots,a_m )=\det[K(a_i

Figures (13)

  • Figure 1: Left: The Young diagram for $\lambda=(6,6,5,3,1,1)$ in the coordinate system rotated by $45^\circ$. The diagonal line $v=u+2N$ represents the upper boundary of the shape of $\lambda$. Right: An example of a limit shape $\mathfrak{W}(u)$ of the Grothendieck random partition for $x=1/3$, $y=1/5$, and $\beta=-25$. We added a horizontal line to highlight the staircase frozen facet where the limit shape $\mathfrak{W}(u)$ is horizontal. An exact sample of a random partition corresponding to the limit shape on the right is given in \ref{['fig:samples']}, right (see also \ref{['sub:sampling']} for a discussion of how to sample Grothendieck random partitions).
  • Figure 2: The directed graph with vertices $\mathbb{Z}_{\ge0}\times\left\{ 1,\ldots,N \right\}$ and edges which can be vertical (with weight $1$) or diagonal (with weight $-\beta_m$, $m=1,\ldots,N$). We consider an ensemble of $N$ nonintersecting paths connecting $k_1,\ldots,k_N$ to $k_1',\ldots,k_N'$. The particles $x^m_j$ encode the intersections of the paths with the $m$-th horizontal line, $m=1,\ldots,N$.
  • Figure 3: Graphical representation of the Schur process \ref{['eq:Schur_process_from_Grothendieck']}. Arrows indicate the diagram inclusion relations.
  • Figure 4: An example of the frozen boundary curve in the $(\xi,\tau)$ coordinates, and an example of several up-diagonal paths as in \ref{['fig:nonintersecting_path_ensemble']} serving as the level lines for the pre-limit height function $H_N$\ref{['eq:prelimit_height_function']}. There are no up-diagonal paths in the frozen zones Ia-b, so $\nabla\mathfrak{H}=(0,0)$. In zone II, the paths go diagonally, so $\nabla\mathfrak{H}=(-1,-1)$. Finally, in zone III, the paths go vertically, so $\nabla\mathfrak{H}=(-1,0)$. These frozen zone gradients correspond to the vertices of the triangle \ref{['eq:gradient_triangle_condition']}. In this example, we have $x=1/3$, $y=1/5$, $\beta=-6$. For other values of parameters, zones Ib and II may be present. Zone III is always present, see \ref{['lemma:uniquely']} below.
  • Figure 5: Triangle in the complex plane with vertices $0,\beta$, and $z_c$. When $z_c$ approaches the real line at $(0,+\infty)$, $(\beta,0)$, and $(-\infty,\beta)$, we have, respectively, $\nabla\mathfrak{H}=(0,0)$, $\nabla\mathfrak{H}=(-1,-1)$, and $\nabla\mathfrak{H}=(-1,0)$.
  • ...and 8 more figures

Theorems & Definitions (40)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3: Application to the five-vertex model
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof : Proof of \ref{['thm:properties_of_W_2d']}
  • Proposition 2.4
  • ...and 30 more