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Gaussian limit for Pfaffian point processes

Kai Wang, Mei Xu

TL;DR

This work proves a central limit theorem for linear statistics of a broad class of Pfaffian point processes by introducing the finite-rank commutator property (FRCP), which captures tractable symmetry in the kernel. The authors extend the moment/cumulant method from determinantal processes to Pfaffian settings and show that high-order cumulants vanish under suitable variance growth, yielding Gaussian limits. The results are applied to the Pfaffian Sine$_4$ and Sine$_1$ processes, where counts in growing intervals exhibit asymptotic normality with logarithmic variance growth, and to step-function statistics via FRCP and trace-class operator techniques. These findings broaden CLTs for Pfaffian ensembles, with implications for GOE/GSE-type spectra and related combinatorial models.

Abstract

We prove a central limit theorem for linear statistics of a broad class of Pfaffian point processes. As an application, we derive Gaussian limits for scaled linear statistics of step functions in the Pfaffian $\mathrm{Sine_4}$ and $\mathrm{Sine}_1$ processes.

Gaussian limit for Pfaffian point processes

TL;DR

This work proves a central limit theorem for linear statistics of a broad class of Pfaffian point processes by introducing the finite-rank commutator property (FRCP), which captures tractable symmetry in the kernel. The authors extend the moment/cumulant method from determinantal processes to Pfaffian settings and show that high-order cumulants vanish under suitable variance growth, yielding Gaussian limits. The results are applied to the Pfaffian Sine and Sine processes, where counts in growing intervals exhibit asymptotic normality with logarithmic variance growth, and to step-function statistics via FRCP and trace-class operator techniques. These findings broaden CLTs for Pfaffian ensembles, with implications for GOE/GSE-type spectra and related combinatorial models.

Abstract

We prove a central limit theorem for linear statistics of a broad class of Pfaffian point processes. As an application, we derive Gaussian limits for scaled linear statistics of step functions in the Pfaffian and processes.

Paper Structure

This paper contains 6 sections, 13 theorems, 158 equations.

Table of Contents

  1. introduction
  2. proof of theorem \ref{['Th2']}
  3. CLT for the Pfaffian $\mathrm{Sine}_4$ and $\mathrm{Sine}_1$ processes
  4. The Pfaffian $\mathrm{Sine}_4$-process
  5. The Pfaffian $\mathrm{Sine}_1$ process
  6. CLT for step functions

Key Result

Theorem 1.2

Fixed an integer $N$. Consider a family $\{\mathbb{P}_L\}_{L\geqslant 0}$ be a family of Pfaffian point processes having FRCP characterized by the data $\{N,f^{(i)}_{L},g^{(i)}_{L},h^{(i)}_{L},e^{(i)}_{L},\alpha_L,\beta_L\}$ with the matrix-valued kernel defined on the domain $X_L$ with respect to the measure $\mu_L$. Under the following conditions: then the normalized counting statistic conver

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 14 more