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Universal Cusp Scaling in Random Partitions

Taro Kimura, Ali Zahabi

Abstract

We study the universal scaling limit of random partitions obeying the Schur measure. Extending our previous analysis [arXiv:2012.06424], we obtain the higher-order Pearcey kernel describing the multi-critical behavior in the cusp scaling limit. We explore the gap probability associated with the higher Pearcey kernel, and derive the coupled nonlinear differential equation and the asymptotic behavior in the large gap limit.

Universal Cusp Scaling in Random Partitions

Abstract

We study the universal scaling limit of random partitions obeying the Schur measure. Extending our previous analysis [arXiv:2012.06424], we obtain the higher-order Pearcey kernel describing the multi-critical behavior in the cusp scaling limit. We explore the gap probability associated with the higher Pearcey kernel, and derive the coupled nonlinear differential equation and the asymptotic behavior in the large gap limit.
Paper Structure (44 sections, 40 theorems, 264 equations)

This paper contains 44 sections, 40 theorems, 264 equations.

Table of Contents

  1. Introduction and summary
  2. Schur measure
  3. Scaling limit
  4. Wave functions
  5. Inclusion of subleading terms
  6. Higher Airy functions
  7. Differential equations
  8. Asymptotic behavior: $\operatorname{Ai}_p$
  9. Positive $z$ regime
  10. Negative $z$ regime
  11. Asymptotic behavior: $\widetilde{\operatorname{Ai}}_p$
  12. Higher Airy kernel
  13. Christoffel--Darboux-type formula
  14. Density function
  15. Even case: $p = 2n$
  16. ...and 29 more sections

Key Result

Proposition 1.1

We have the following asymptotic behavior of the Schur measure kernel under the multicritical condition eq:multicrit_cond, where $(\alpha_p, \beta)$ are the constants defined in eq:alpha_beta, and $K_{p\text{-Airy}}(\cdot,\cdot)$ is the higher Airy kernel constructed with the higher Airy functions (Definition def:higher_Airy_kernel).

Theorems & Definitions (88)

  • Proposition 1.1: Proposition \ref{['prop:kernel_scaling_lim']}
  • Proposition 1.2: Chistoffel--Darboux-type formula, Proposition \ref{['prop:CD_formula']}
  • Proposition 1.3: Proposition \ref{['prop:large_gap']}
  • Definition 2.1: Schur measure
  • Definition 2.2: Correlation function
  • Definition 2.3: Wave functions
  • Remark 2.4
  • Proposition 2.5: Determinantal formula Okounkov:2001SM
  • Lemma 2.6
  • proof
  • ...and 78 more