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Global asymptotics for $β$-Krawtchouk corners processes via multi-level loop equations

Evgeni Dimitrov, Alisa Knizel

Abstract

We introduce a two-parameter family of probability distributions, indexed by $β/2 = θ> 0$ and $K \in \mathbb{Z}_{\geq 0}$, that are called $β$-Krawtchouk corners processes. These measures are related to Jack symmetric functions, and can be thought of as integrable discretizations of $β$-corners processes from random matrix theory, or alternatively as non-determinantal measures on lozenge tilings of infinite domains. We show that as $K$ tends to infinity the height function of these models concentrates around an explicit limit shape, and prove that its fluctuations are asymptotically described by a pull-back of the Gaussian free field, which agrees with the one for Wigner matrices. The main tools we use to establish our results are certain multi-level loop equations introduced in our earlier work arXiv:2108.07710.

Global asymptotics for $β$-Krawtchouk corners processes via multi-level loop equations

Abstract

We introduce a two-parameter family of probability distributions, indexed by and , that are called -Krawtchouk corners processes. These measures are related to Jack symmetric functions, and can be thought of as integrable discretizations of -corners processes from random matrix theory, or alternatively as non-determinantal measures on lozenge tilings of infinite domains. We show that as tends to infinity the height function of these models concentrates around an explicit limit shape, and prove that its fluctuations are asymptotically described by a pull-back of the Gaussian free field, which agrees with the one for Wigner matrices. The main tools we use to establish our results are certain multi-level loop equations introduced in our earlier work arXiv:2108.07710.
Paper Structure (35 sections, 24 theorems, 533 equations, 5 figures)

This paper contains 35 sections, 24 theorems, 533 equations, 5 figures.

Table of Contents

  1. Introduction and main result
  2. Preface
  3. Lozenge tilings
  4. General model
  5. Main results and methods
  6. Paper outline
  7. Connection to Jack symmetric functions
  8. Jack symmetric functions
  9. Ascending Jack processes
  10. From $\mathcal{J}^{\theta, K}_N$ to ${\mathbb{P}^{\theta, K}_{N}}$
  11. Single-level analysis
  12. Law of large numbers
  13. Weak moment bounds
  14. Strong moment bounds
  15. Two-level analysis
  16. ...and 20 more sections

Key Result

Theorem 1.4

Fix $\theta > 0$, assume that $(\ell^1, \ell^2, \dots)$ is distributed according to $\mathbb{P}^{\theta, K}_{\infty}$ as in Definition BKCC and let $\mathcal{H}_K$ be as in (S1DHF). Then, for any compact set $\mathcal{V} \subset \mathbb{R} \times (0,\infty)$ we have where $h(x,s)$ is as in (S1LimHeight) and the convergence is in probability.

Figures (5)

  • Figure 1: The left part depicts the three lozenges and a tiling of a domain. The gray region is a sub-domain and the conditional distribution of its tiling is uniform over all three tilings shown on the right.
  • Figure 2: The left part depicts a trapezoidal domain. The right part depicts a random tiling of a trapezoid with base $K =5$ and height $N = 4$. For the right side, the particle locations on the fourth level are $\ell_1^4 = 2$, $\ell_2^4 = -1$, $\ell_3^4 = -3$ and $\ell^4_4 = -5$.
  • Figure 3: The liquid region $\mathcal{D}$ and the map $\Omega$ that sends it to $\mathbb{H}$.
  • Figure 4: The contour $\gamma_{\varepsilon} = \cup_{i = 1}^4 \gamma_{\varepsilon}^i$.
  • Figure 5: The contour $\Gamma_{\varepsilon, \epsilon}$.

Theorems & Definitions (75)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Remark 2.2
  • ...and 65 more