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

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.