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The 2D Toda lattice hierarchy for multiplicative statistics of Schur measures

Pierre Lazag

TL;DR

The paper shows that Fredholm determinants built from multiplicative statistics of Schur measures are $ au$-functions of the $2$D Toda lattice hierarchy. It extends Okounkov’s and Cafasso–Ruzza’s results to arbitrary multiplicative statistics, including finite-temperature Schur measures, by expressing the determinants as expectations under Schur measures and realizing them within the semi-infinite wedge formalism via the Boson-Fermion correspondence. The main contribution is a uniform integrable structure for a broad class of deformations of Schur measures, captured by the bilinear Hirota equations. This framework links determinantal point processes to integrable hierarchies and paves the way for further connections to finite-temperature kernels and related Painlevé-type systems.

Abstract

We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative statistics of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to finite temperature Schur measures, and extends both the result of Okounkov in \cite{okounkovschurmeasures} and of Cafasso-Ruzza in \cite{cafassoruzza} concerning the finite-temperature Plancherel measure. Our proof lies on the semi-infinite wedge formalism and the Boson-Fermion correspondance.

The 2D Toda lattice hierarchy for multiplicative statistics of Schur measures

TL;DR

The paper shows that Fredholm determinants built from multiplicative statistics of Schur measures are -functions of the D Toda lattice hierarchy. It extends Okounkov’s and Cafasso–Ruzza’s results to arbitrary multiplicative statistics, including finite-temperature Schur measures, by expressing the determinants as expectations under Schur measures and realizing them within the semi-infinite wedge formalism via the Boson-Fermion correspondence. The main contribution is a uniform integrable structure for a broad class of deformations of Schur measures, captured by the bilinear Hirota equations. This framework links determinantal point processes to integrable hierarchies and paves the way for further connections to finite-temperature kernels and related Painlevé-type systems.

Abstract

We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative statistics of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to finite temperature Schur measures, and extends both the result of Okounkov in \cite{okounkovschurmeasures} and of Cafasso-Ruzza in \cite{cafassoruzza} concerning the finite-temperature Plancherel measure. Our proof lies on the semi-infinite wedge formalism and the Boson-Fermion correspondance.
Paper Structure (12 sections, 8 theorems, 77 equations)

This paper contains 12 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Proof's strategy and organisation of the paper
  4. On Schur measures and finite temperature Schur measures
  5. Schur measures
  6. Finite temperature Schur measures
  7. Schur measures and finite temperature Schur measures as determinantal point processes
  8. Semi-infinite wedge formalism
  9. Definition
  10. Operators on $\Lambda^{\frac{\infty}{2}}V$
  11. Intermediate results
  12. Proof of Theorem \ref{['thm:thm1']}

Key Result

Proposition 1.1

The restriction of $K_{t,t',\sigma}$ on $l^2\{ n +1/2,n+3/2,\dots \}$ is trace class for any $n \in \mathbb{Z}$.

Theorems & Definitions (14)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.1
  • Proposition 3.2
  • ...and 4 more