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Register a new userLarge deviations for the largest eigenvalue of an Hermitian Brownian motion
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Authors
Catherine Donati-Martin
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We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.
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In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.
The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
In this article, we study high-dimensional behavior of empirical spectral distributions ${L_N(t), tin[0,T]}$ for a class of $Ntimes N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H in(1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of ${L_N(t), tin[0,T]}_{Ninmathbb N}$ in $C([0,T], mathbf P(mathbb R))$ following the approach in cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of ${L_N(t), tin[0,T]}_{Ninmathbb N}$ on $C([0,T], mathbf P(mathbb R))$ by tightness criterions provided in Appendix ref{subset:tightness argument}. The limit of ${L_N(t), tin[0,T]}$ as $Nto infty$ is also characterised.
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 $X_s^{(max)}stackrel{D}{=} sqrt{4s}lambda_{max}$.
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