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Register a new userAsymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble
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Consider a $n times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(Delta_{i,n}, 1leq ileq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigners semi-circle law, we prove that the related counting measures $({mathcal N}_n(Delta_{i,n}), 1leq ileq p)$, where ${mathcal N}_n(Delta)$ represents the number of eigenvalues within $Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.
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We study stochastic perturbations of ODE with stable limit cycles -- referred to as stochastic oscillators -- and investigate the response of the asymptotic (in time) frequency of oscillations to changing noise amplitude. Unlike previous studies, we do not restrict our attention to the small noise limit, and account for the fact that large deviation events may push the system out of its oscillatory regime. To do so, we consider stochastic oscillators conditioned on their remaining in an oscillatory regime for all time. This leads us to use the theory of quasi-ergodic measures, and to define quasi-asymptotic frequencies as conditional, long-time average frequencies. We show that quasi-asymptotic frequencies always exist, though they may or may not be observable in practice. Our discussion recovers previous results on stochastic oscillators in the literature. In particular, existing results imply that the asymptotic frequency of a stochastic oscillator depends quadratically on the noise amplitude. We describe scenarios where this prediction holds, though we also show that it is not true in general -- even for small noise.
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakais geometric theory of Painleve equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painleve equations.
We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langers generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.
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 measure on the unit disc, both in expectation and almost surely. For $1 le p le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.
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