We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.Comment: Published in at http://dx.doi.org/10.1214/13-AAP958 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We give a method for computing the ensemble average of multiplicative class functions over the Gaussian ensemble of real asymmetric matrices. These averages are expressed in terms of the Pfaffian of Gram-like antisymmetric matrices formed with respect to a skew-symmetric inner product related to the class function.

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2 : 1 in the presence of the harmonic oscillator potential. The system is assumed to be at temperature corresponding to β = 1 and the sum of the charges is fixed. We investigate the distribution of the number as well as the spatial density of each species of particle in the limit as the total charge increases to ∞. These results will follow from the fact that the system of particles forms a Pfaffian point process. We produce the skew-orthogonal polynomials necessary to simplify the related matrix kernels.

Using the method of Tracy and Widom we rederive the correlation functions for β = 1 Hermitian and real asymmetric ensembles of N × N matrices with N odd.