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Dynamical Loop Equation

Vadim Gorin, Jiaoyang Huang

Abstract

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two--dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, $β$--corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and $(q,κ)$-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for $(q,κ)$--distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.

Dynamical Loop Equation

Abstract

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two--dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, --corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and -distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for --distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.
Paper Structure (40 sections, 50 theorems, 487 equations, 10 figures)

This paper contains 40 sections, 50 theorems, 487 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Dynamical loop equations
  4. Height fluctuations of $(q,\kappa)$-distributions on lozenge tilings
  5. Dynamical Loop Equation
  6. Proof of Theorem \ref{['t:loopeq']}
  7. Particle-Hole Duality
  8. Examples of Dynamical Loop Equations
  9. Nonintersecting Bernoulli random walks
  10. Nonintersecting Poisson random walks
  11. Dyson Brownian motion
  12. Measures on Gelfand-Tsetlin Patterns
  13. $\beta$--Corners processes
  14. Macdonald processes
  15. Macdonald polynomials and specializations
  16. ...and 25 more sections

Key Result

Theorem 1.1

Choose an open set $U\subset \mathbb C$, a particle configuration ${{\bm{x}}=(x_1>x_2>\dots>x_n)\in \mathbb{W}_\theta^n}$ such that the interval $[x_n,x_1]\subset U$, a parameter $\theta>0$, two holomorphic functions $\phi^+(z)$, $\phi^-(z)$ on $U$ and a conformal (i.e., holomorphic and injective) f

Figures (10)

  • Figure 1: We identify $(\lambda_1,\lambda_2,\dots,\lambda_n)\in \mathbb{GT}_n$ with $n$-particles $(\lambda_1,\lambda_2-\theta,\lambda_3-2\theta,\dots,\lambda_n-(n-1)\theta)$.
  • Figure 2: Left panel: trapezoid domain with $N=8$, $T=12$, $a_1=-6$, $b_1=-4$, $a_2=-2$, $b_2=4$; and $(t,x)$--coordinate system. Middle panel: one possible lozenge tiling and the values for $\tilde{x}$ in \ref{['eq_lozenge_weight']}. Right panel: non-intersecting paths and values of the height function.
  • Figure 3: Samples of tilings of $100\times 100 \times 100$ hexagon. Left panel: imaginary case with $\kappa=\infty$, $q=0.97$. Center panel: real case with $q=0.99$, $\kappa=q^{-110}$. Right panel: trigonometric case with $\alpha=0.015$, $\beta=1.57$.
  • Figure 4: Arctic curves via Theorem \ref{['t:arctic']}. Left panel: ${\mathsf q}=\tfrac{1}{10}$, $\kappa=\frac{\mathbf i}{10}$. Right panel: ${\mathsf q}=20$, $\kappa= \frac{\mathbf i}{10}$.
  • Figure 5: Triangle $(0,1,f)$ has angles $(\pi p_{ {}}, \pi p_{ {\hbox{,}}},\pi p_{ {\hbox{,}}})$.
  • ...and 5 more figures

Theorems & Definitions (133)

  • Theorem 1.1: Dynamical Loop Equation
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 1.9
  • Remark 1.10
  • ...and 123 more