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Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functions

Alexander I. Bufetov

TL;DR

The paper addresses sub-Poissonian tail behavior for the number of particles in an interval under determinantal and Pfaffian point processes with kernels that are restrictions of entire functions of finite order. It develops a mechanism based on divided-differences bounds for determinants/Pfaffians and standard derivative estimates of entire functions to bound the probability of large counts and to control exponential moments of the squared count, yielding E e^{\\lambda \\\#_I^2} ≤ exp( c ( e^{\\lambda\\sigma} - 1 ) ) where σ is the kernel order. A corollary extends these moment bounds to additive functionals over pairs of particles, via a uniform kernel assumption and the norm ||q||_{(1,∞)}. The approach is applied to classical kernels (sine, Bessel, Airy) and generalized Fock spaces, producing explicit constants and illuminating the role of kernel order, with discussion on limitations for Pfaffian cases and potential extensions.

Abstract

The aim of this note is to estimate the tail of the distribution of the number of particles in an interval under determinantal and Pfaffian point processes. The main result of the note is that the square of the number of particles under the determinantal point process whose correlation kernel is an entire function of finite order has sub-Poissonian tails. The same result also holds in the symplectic Pfaffian case. As a corollary, sub-Poissonian estimates are also obtained for exponential moments of additive functionals over pairs of particles.

Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functions

TL;DR

The paper addresses sub-Poissonian tail behavior for the number of particles in an interval under determinantal and Pfaffian point processes with kernels that are restrictions of entire functions of finite order. It develops a mechanism based on divided-differences bounds for determinants/Pfaffians and standard derivative estimates of entire functions to bound the probability of large counts and to control exponential moments of the squared count, yielding E e^{\\lambda \\\#_I^2} ≤ exp( c ( e^{\\lambda\\sigma} - 1 ) ) where σ is the kernel order. A corollary extends these moment bounds to additive functionals over pairs of particles, via a uniform kernel assumption and the norm ||q||_{(1,∞)}. The approach is applied to classical kernels (sine, Bessel, Airy) and generalized Fock spaces, producing explicit constants and illuminating the role of kernel order, with discussion on limitations for Pfaffian cases and potential extensions.

Abstract

The aim of this note is to estimate the tail of the distribution of the number of particles in an interval under determinantal and Pfaffian point processes. The main result of the note is that the square of the number of particles under the determinantal point process whose correlation kernel is an entire function of finite order has sub-Poissonian tails. The same result also holds in the symplectic Pfaffian case. As a corollary, sub-Poissonian estimates are also obtained for exponential moments of additive functionals over pairs of particles.
Paper Structure (10 sections, 19 theorems, 94 equations)

This paper contains 10 sections, 19 theorems, 94 equations.

Table of Contents

  1. Introduction and the formulation of the main result
  2. Acknowledgements.
  3. Beginning of the proof of Theorem \ref{['thm:main']}: divided differences and estimates for determinants
  4. Estimates of entire functions
  5. Conclusion of the proof of Theorem \ref{['thm:main']}
  6. The case of the sine-process
  7. The case of the Bessel kernel
  8. The case of the Airy kernel
  9. The case of the generalized Fock kernel
  10. Concluding remarks

Key Result

Theorem 1

There exist positive constants $B$ and $c$ depending only on $A$, $M$, $\sigma$, $|I|$, and $\sup_I\rho$ such that under Assumption asm:1 the number of particles $\#_I$ in the interval $I$ satisfies, under the determinantal point process ${\mathbb P}_\Pi$, the estimates and

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor:1']}.
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 20 more