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Register a new userA non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
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We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $Ntimes (N+ u)$ are correlated through a non-Hermiticity parameter $tauin[0,1]$. For general $ u=O(N)$ and $tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $mathcal{D}=begin{pmatrix} 0 & X_1 X_2^* & 0 end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostmans equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
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In random matrix theory, Marchenko-Pastur law states that random matrices with independent and identically distributed entries have a universal asymptotic eigenvalue distribution under large dimension limit, regardless of the choice of entry distribution. This law provides useful insight for physics research, because the large $N$ limit proved to be a very useful tool in various theoretical models. We present an alternative proof of Marchenko- Pastur law using Feynman diagrams, which is more familiar to the physics community. We also show that our direct diagrammatic approach can readily generalize to six types of restricted random matrices, which are not all covered by the original Marchenko-Pastur law.
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ e}$ for non-Hermitian random matrices at any point $z in C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $ |z|-1=oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $ |z|-1=oo(1)$ up to scale $N^{-1/4+ e}$.
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.
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.
Recently Johansson and Rahman obtained the limiting multi-time distribution for the discrete polynuclear growth model, which is equivalent to discrete TASEP model with step initial condition. In this paper, we obtain a finite time multi-point distribution formula of continuous TASEP with general initial conditions in the space-time plane. We evaluate the limit of this distribution function when the times go to infinity proportionally for both step and flat initial conditions. These limiting distributions are expected to be universal for all the models in the Kardar-Parisi-Zhang universality class.
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