On the distribution of the largest real eigenvalue for the real Ginibre ensemble

DOI10.1214/16-AAP1233zbMATH Open1375.60023arXiv1603.05849MaRDI QIDQ2403131

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Publication date: 15 September 2017

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Abstract: 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'). We study the large deviations behaviour of the limiting $Nightarrowinfty$ distribution $P[lambd{a}_{max}<t]$ of the shifted maximal real eigenvalue $lambd{a}_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, [ P[lambda_{max}<t]=1-frac{1}{4}mbox{erfc}(t)+Oleft(e^{-2t^2} ight). ] This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: for $t<0$, [ P[lambda_{max}<t]= e^{frac{1}{2sqrt{2pi}}zetaleft(frac{3}{2} ight)t+O(1)}, ] where $zeta$ is the Riemann zeta-function. Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABM's) with the step initial condition. Therefore, the tail behaviour of the distribution of $X_s^{(max)}$ - the position of the rightmost annihilating particle at fixed time $s>0$ - can be read off from the corresponding answers for $lambd{a}_{max}$ using $X_s^{(max)}stackrelD=sqrt4slambd{a}_{max}$.

Full work available at URL: https://arxiv.org/abs/1603.05849

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