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Multi-level loop equations for $β$-corners processes

Evgeni Dimitrov, Alisa Knizel

Abstract

The goal of the paper is to introduce a new set of tools for the study of discrete and continuous $β$-corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger-Dyson equations) for $β$-log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete $β$-ensembles obtained by Borodin, Gorin and Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).

Multi-level loop equations for $β$-corners processes

Abstract

The goal of the paper is to introduce a new set of tools for the study of discrete and continuous -corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger-Dyson equations) for -log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete -ensembles obtained by Borodin, Gorin and Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).

Paper Structure

This paper contains 17 sections, 9 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Preface
  3. Main results
  4. Discrete setting
  5. Continuous setting
  6. General setup and the equations
  7. Preliminary computations
  8. Multi-level Nekrasov equations: $\theta \neq 1$
  9. Multi-level Nekrasov equations: $\theta = 1$
  10. Application of Nekrasov equations
  11. Joint cumulants
  12. Deformed measures
  13. Continuous limits
  14. Continuous $\beta$-corners processes
  15. Jack symmetric functions
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Let ${\mathbb{P}^{\theta,M}_{N,k}}$ be a measure as in (S1PDef) for $\theta > 0$, $\theta \neq 1$, $N \in \mathbb{N}$, $k \in \llbracket 1, N \rrbracket$, $M \in \mathbb{Z}_{\geq 0}$. Let $\mathcal{M} \subseteq \mathbb{C}$ be an open set and $[- N \cdot \theta, M + 1 - \theta] \subseteq \mathcal{M}$ Then the following functions $R_1(z)$, $R_2(z)$ are analytic in $\mathcal{M}$:

Figures (2)

  • Figure 1: The figure depicts the action of the map $\mathfrak{b}^{i,s}_1$. To make the action comprehensible we explain how it works in the changed coordinates $x_i^j = \ell_i^j + i \cdot (\theta -1)$ -- this way the $x$'s all lie on the integer lattice and $x_1^j, \dots, x_j^j$ are distinct. The function $\mathfrak{b}^{i,s}_1$ takes as input $(\ell,n)$ and in the changed coordinates looks at the longest string of particles $x^j_i$ for $k \leq j \leq n$ that have the same horizontal coordinate as $x_i^n$. This string is denoted by $x^{\tilde{n}}_i, \dots,x^{n}_i$ and $\mathfrak{b}^{i,s}_1$ acts by shifting all these particles to the left by one. In the picture $N = n = 4$, $k = \tilde{n} = 2$ and $i = 1$.
  • Figure 2: The figure depicts the action of the map $\mathfrak{b}^{i,s}_1$. To make the action comprehensible we explain how it works in the changed coordinates $x_i^j = \ell_i^j + i \cdot (\theta -1)$ -- this way the $x$'s all lie on the integer lattice and $x_1^j, \dots, x_j^j$ are distinct. The function $\mathfrak{b}^{i,s}_2$ takes as input $(\ell,n)$ and in the changed coordinates looks at the longest string of particles $x^j_{j -i + 1}$ for $N \geq j \geq n$ that lie on the line of slope $-1$ and above $x^n_{n-i+1}$. This string is denoted by $x^{n}_{n- i + 1}, \dots,x^{\tilde{n}}_{\tilde{n} - i+1}$ and $\mathfrak{b}^{i,s}_2$ acts by shifting all these particles to the left by one. In the picture we have $N = 4$, $k = n = 2$, $\tilde{n} = 3$ and $i = 2$.

Theorems & Definitions (29)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 19 more