Sharp asymptotics for Fredholm pfaffians related to interacting particle systems and random matrices

DOI10.1214/20-EJP512zbMATH Open1462.60007arXiv1905.03754MaRDI QIDQ2024499

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Publication date: 4 May 2021

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Abstract: It has been known since the pioneering paper of Mark Kac, 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 $sqrtN+lambd{a}_{max}$ be the largest real eigenvalue of a random $NimesN$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). Consider the limiting $Nightarrowinfty$ distribution $mathbbP[lambd{a}_{max}<-L]$ of the shifted maximal real eigenvalue $lambd{a}_{max}$. Then [ lim_{L ightarrow infty} e^{frac{1}{2sqrt{2pi}}zetaleft(frac{3}{2} ight)L} mathbb{P}left(lambda_{max}<-L ight) =e^{C_e}, ] where $zeta$ is the Riemann zeta-function and [ C_e=frac{1}{2}log 2+frac{1}{4pi}sum_{n=1}^{infty}frac{1}{n} left(-pi+sum_{m=1}^{n-1}frac{1}{sqrt{m(n-m)}} ight). ] 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 ightarrow infty} e^{frac{1}{2sqrt{2pi}}zetaleft(frac{3}{2} ight)L} mathbb{P}left(frac{X_{t}^{(max)}}{sqrt{4t}}<-L ight) =e^{C_e}. ] These statements are a sharp counterpart of our previous results improved by computing the terms of order $L^0$ in the asymptotic expansion of the corresponding Fredholm Pfaffian.

Full work available at URL: https://arxiv.org/abs/1905.03754

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