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Let $sqrt{N}+lambda_{max}$ be the largest real eigenvalue of a random $Ntimes N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix). We study the large deviations behaviour of the limiting $Nrightarrow infty$ distribution $P[lambda_{max}<t]$ of the shifted maximal real eigenvalue $lambda_{max}$. 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}right). ] 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}right)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 (ABMs) 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 $lambda_{max}$ using $X_s^{(max)}stackrel{D}{=} sqrt{4s}lambda_{max}$.
We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform meas
We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the c
Let $G_n$ be an $n times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n times n$ matrix with $Vert A Vert le 1$, and let $gamma in (0,1)$. We show that with probability $0.99$, $A + gamma G_n$ has all of its eigenvalue condition
Nowadays, strict finite size effects must be taken into account in condensed matter problems when treated through models based on lattices or graphs. On the other hand, the cases of directed bonds or links are known as highly relevant, in topics rang
We consider the real eigenvalues of an $(N times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $tau_Nin [0,1]$. In the almost-Hermitian regime where $1-tau_N=Theta(N^{-1})$, we obtain the large-$N$