The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s (2N)1 2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble an...
The paper @cite_19 by Ben Arous, Baik and Pech 'e proved the phase transition property relating to ) in the complex case with @math given by @math () corresponds to @math ). Subsequent studies by Baik and Silverstein @cite_31 , Paul @cite_7 and Bai and Yao @cite_15 considered the real case. Significant for the present study is the result of @cite_15 , giving that for ) with @math given as above, the separated eigenvalues have the law of the @math GUE. The case @math ---not considered here---corresponding to self dual quaternion real matrices, is studied in the recent work of Wang @cite_39 .
@cite_7 @cite_15 @cite_39 @cite_19 @cite_31
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Let (a>0, b>0, ab<1; ) and let (g L^2( R ). ) In this paper we investigate the relation between the frame operator (S:f L^2( R ) n,m ,(f,g_ na,mb ) ,g_ na,mb ) and the matrix (H ) whose entries (H_ k,l ,; ,k',l' ) are given by ((g_ k' b,l' a ,g_ k b,l a ) ) for (k,l,k',l' Z . ) Here (f_ x,y (t)= exp (2 iyt) ,f(t-x), ) (t R ), for any (f L^2( R ). ) We show that (S ) is bounded as a mapping of (L^2( R ) ) into (L^2( R ) ) if and only if (H ) is bounded as a mapping of (l^2( Z ^2) ) into (l^2( Z ^2). ) Also we show that (AI S BI ) if and only if (AI 1 ab ,H BI, ) where (I ) denotes the identity operator of (L^2( R ) ) and (l^2( Z ^2), ) respectively, and (A 0, ) (B< . ) Next, when (g ) generates a frame, we have that ((g_ k b,l a )_ k,l ) has an upper frame bound, and the minimal dual function (^ ) can be computed as (ab , k,l ,(H^ -1 )_ k,l ,; ,o,o ,g_ k b,l a . ) The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a (g ) generating a frame are inherited by (^ . ) In particular, we show that (^ S ) when (g S ) generates a frame (( S ) Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Abstract Let A be a d × n matrix and T = Tn -1 be the standard simplex in R n. Suppose that d and n are both large and comparable: d ≈ δn, δ ∈ (0, 1). We count the faces of the projected simplex AT when the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R n. We derive ρ N(δ) > 0 with the property that, for any ρ 0 at which phase transition occurs in k d. We compute and display ρ VS and compare with ρ N. Corollaries are as follows. (1) The convex hull of n Gaussian samples in Rd , with n large and proportional to d, has the same k-skeleton as the (n - 1) simplex, for k < ρ N (d n)d(1 + oP (1)). (2) There is a “phase transition” in the ability of linear programming to find the sparsest nonnegative solution to systems of underdetermined linear equations. For most systems having a solution with fewer than ρ VS (d n)d(1 + o(1)) nonzeros, linear programming will find that solution. neighborly polytopes convex hull of Gaussian sample underdetermined systems of linear equations uniformly distributed random projections phase transitions Let A be a complex matrix with arbitrary Jordan structure and @math an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S. R. Comput. Math. and Math. Phys., 1 (1965), pp. 73--85], showing that the splitting of @math under a small perturbation of A of order @math is, generically, of order @math . Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Holder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors. Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004].