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2020 Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices
Will FitzGerald, Roger Tribe, Oleg Zaboronski
Electron. J. Probab. 25: 1-15 (2020). DOI: 10.1214/20-EJP512

## Abstract

It has been known since the pioneering paper of Mark Kac [20], that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac’ approach by studying the Fredholm Pfaffian describing the statistics of both non-Hermitian random matrices and annihilating Brownian motions. Namely, we establish the following two results. Firstly, let $\sqrt {N}+\lambda _{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the ‘real Ginibre matrix’). Consider the limiting $N\rightarrow \infty$ distribution $\mathbb {P}[\lambda _{max}<-L]$ of the shifted maximal real eigenvalue $\lambda _{max}$. Then $\lim _{L\rightarrow \infty } e^{\frac {1}{2\sqrt {2\pi }}\zeta \left (\frac {3}{2}\right )L} \mathbb {P}\left (\lambda _{max}<-L\right ) =e^{C_{e}},$ where $\zeta$ is the Riemann zeta-function and $C_{e}=\frac {1}{2}\log 2+\frac {1}{4\pi }\sum _{n=1}^{\infty }\frac {1}{n} \left (-\pi +\sum _{m=1}^{n-1}\frac {1}{\sqrt {m(n-m)}}\right ).$ Secondly, let $X_{t}^{(max)}$ be the position of the rightmost particle at time $t$ for a system of annihilating Brownian motions (ABM’s) started from every point of $\mathbb {R}_{-}$. Then $\lim _{L\rightarrow \infty } e^{\frac {1}{2\sqrt {2\pi }}\zeta \left (\frac {3}{2}\right )L} \mathbb {P}\left (\frac {X_{t}^{(max)}}{\sqrt {4t}}<-L\right ) =e^{C_{e}}.$ These statements are a sharp counterpart of the results of [22], improved by computing the $O(L^{0})$ term in the asymptotic $L\rightarrow \infty$ expansion of the corresponding Fredholm Pfaffian.

## Citation

Will FitzGerald. Roger Tribe. Oleg Zaboronski. "Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices." Electron. J. Probab. 25 1 - 15, 2020. https://doi.org/10.1214/20-EJP512

## Information

Received: 14 May 2019; Accepted: 26 August 2020; Published: 2020
First available in Project Euclid: 25 September 2020

MathSciNet: MR4161126
Digital Object Identifier: 10.1214/20-EJP512

Subjects:
Primary: 60B20, 82C22

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Vol.25 • 2020