Random matrix products: Universality and least singular values

Singular Values and Least Squares Matching

In this paper we investigate least squares matching problems, extending the methods of our earlier paper 1] in such a way as to make them applicable to problems involving the sets of points that are so complex that approximate answers are of interest. These problems are formulated in terms of continuous descent equations, and lower bounds on the quality of the best match are obtained in terms o...

Universality in Random Matrix Theory

which is the Central Limit Theorem. In principle, all the random variables X1, X2, · · · , XN can be of order 1, hence SN ∼ 1 as well, but the probability of having such a rare event is incredibly small. We can even estimate the bound on the probability for the rare event from the large deviation principle. A similar phenomenon happens when we form a large matrix from i.i.d. random variables an...

\random" Random Matrix Products

Traces of Singular Values and Borcherds Products

Abstract. Let p be a prime for which the congruence group Γ0(p) ∗ is of genus zero, and j∗ p be the corresponding Hauptmodul. Let f be a nearly holomorphic modular form of weight 1/2 on Γ0(4p) which satisfies some congruence condition on its Fourier coefficients. We interpret f as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite produ...

Universality of a double scaling limit near singular edge points in random matrix models

We consider unitary random matrix ensembles Z n,s,te −n tr s,tdM on the space of Hermitian n × n matrices M , where the confining potential Vs,t is such that the limiting mean density of eigenvalues (as n → ∞ and s, t → 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double sc...