Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability p_{n,k} to find exactly k real eigenvalues in the spectrum of an nxn real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined

Hole probabilities and overcrowding estimates for products of complex Gaussian matrices

Akemann G, Strahov E. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. Journal of Statistical Physics. 2013;151(6):987-1003.We consider eigenvalues of a product of n non-Hermitian, independent randommatrices. Each matrix in this product is of size N\times N with independentstandard complex Gaussian variables. The eigenvalues of such a product form adeterminantal point process on the complex plane (Akemann and Burda J. Phys A:Math. Theor. 45 (2012) 465201), which can be understood as a generalization ofthe finite Ginibre ensemble. As N\rightarrow\infty, a generalized infiniteGinibre ensemble arises. We show that the set of absolute values of the pointsof this determinantal process has the same distribution as{R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2is distributed as the product of n independent Gamma variables Gamma(k,1). Thisenables us to find the asymptotics for the hole probabilities, i.e. for theprobabilities of the events that there are no points of the process in a discof radius r with its center at 0, as r\rightarrow\infty. In addition, we solvethe relevant overcrowding problem: we derive an asymptotic formula for theprobability that there are more than m points of the process in a fixed disk ofradius r with its center at 0, as m\rightarrow\infty