It is well known that an @math Wishart matrix with @math degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if @math is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when @math . Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when @math . This shows, in particular, that the phase transition from Wishart to GOE is smooth.
Theorems and state that as @math , statistics of the Wishart @math and the GOE @math have asymptotically the same distribution, but this is not the case if @math remains bounded. In the random matrix literature there has been lots of work showing that statistics of these ensembles have asymptotically the same distribution even when @math . For instance, when @math , then the limiting empirical spectral distribution of the Wishart is the Marchenko-Pastur law, which shows the difference between the Wishart and GOE, but the largest eigenvalue of the Wishart already behaves like that of the GOE @cite_2 @cite_8 @cite_6 . This naturally raises the question of whether there are other regimes of @math and @math where there are interesting phase transitions.
@cite_8 @cite_6 @cite_2
1995012902 2522434885 2081951537 1498064444
We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy–Widom law can be approximated by a properly scaled and shifted gamma distribution, with great accuracy for the values of common interest in statistical applications. A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, in which a prominent eigenvector is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Peche showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the signal strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise. However, not all the information about the spike is necessarily contained in the spectrum. We study the fundamental limitations of statistical methods, including non-spectral ones. Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike. We extend previous work on the spherically symmetric and i.i.d. Rademacher priors through an elementary, unified analysis. II) For any non-Gaussian Wigner ensemble, we show that PCA is always suboptimal for detection. However, a variant of PCA achieves the optimal threshold (for benign priors) by pre-transforming the matrix entries according to a carefully designed function. This approach has been stated before, and we give a rigorous and general analysis. III) For both the Gaussian Wishart ensemble and various synchronization problems over groups, we show that inefficient procedures can work below the threshold where PCA succeeds, whereas no known efficient algorithm achieves this. This conjectural gap between what is statistically possible and what can be done efficiently remains open. Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C^d tensor C^d obtained after partial tracing a random pure state over some ancilla of dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when alpha<4. We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE @math , which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE @math as the square of GOE'', ours has an advantage that it can also describe the transition from the GUE Tracy-Widom distribution to the GOE @math . We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain non-colliding Brownian motion model and the multi-layer PNG model with an external source. This provides us with a multi-matrix model interpretation of the multi-point height distributions of the PNG model with an external source.