2018/08/07 by Baik, Jinho, Bothner, Thomas · 1 citation
#60G70 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 60B20 #Probability (math.PR) #Secondary 45M05
paper · doi:10.48550/arxiv.1808.02419
The real Ginibre ensemble consists of n× n real matrices \bf X whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius Rn=max1≤ j≤ n|zj(\bf X)| of the eigenvalues zj(\bf X)∈ℂ of a real Ginibre matrix \bf X follows a different limiting law (as n→∞) for zj(\bf X)∈ℝ than for zj(\bf X)∈ℂ∖ℝ. Building on previous work by Rider, Sinclair \citeRS and Poplavskyi, Tribe, Zaboronski \citePTZ, we show that the limiting distribution of maxj:zj∈ℝzj(\bf X) admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. As byproducts of our analysis we also obtain a new determinantal representation for the limiting distribution of maxj:zj∈ℝzj(\bf X) and extend recent tail estimates in \citePTZ via nonlinear steepest descent techniques.