We study the empirical measure L A n of the eigenvalues of non-normal square matrices of the form A n = U n D n V n with U n ,V n independent Haar distributed on the unitary group and D n real diagonal. We show that when the empirical measure of the eigenvalues of D n converges, and D n satisfies some technical conditions, L A n converges towards a rotationally invariant measure on the complex ...

Abstract. This note presents some equalities in law for ZN := det(Id− G), where G is an element of a subgroup of the set of unitary matrices of size N , endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways : either by a r...

Take a generic subgroup G, endowed with its Haar measure, from U(n,K), the unitary group of dimension n over the field K of real, complex or quaternion numbers. We give some equalities in law for Z := det(Id − G), G ∈ G : under some general conditions, Z can be decomposed as a product of independent random variables, whose laws are explicitly known (Section 2). Consequently G, endowed with a ge...

Let M be a monoid (e. is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M ?! A M that commutes with all shift maps. Let be a (possibly nonstationary) probability measure on A M ; we develop suucient conditions on and F so that the sequence fF N g 1 N=1 weak*-converges to the Haar measure on A M , in density (and thus, in Cess aro average as we...

The strength of any conventional cipher system relies on the underlying cryptographic Boolean functions employed in the system. The design of such systems requires that the employed Boolean functions meet specific security criteria. Two of such criteria are the correlation immunity and resiliency of a given Boolean function. To determine whether such criteria are met, a designer needs the help ...

Several wavelet synopsis construction algorithms were previously proposed based on dynamic programming for unrestricted Haar wavelet synopses as well as Haar synopses. However, they find an optimal synopsis for every incoming value in each node of a coe cient tree, even if di↵erent incoming values share an identical optimal synopsis. To alleviate the limitation, we present novel algorithms, whi...

A conjecture has previously been made on the chaotic behavior of the eigenvectors of a class of n-dimensional random matrices, where n is very large [ Evidence supporting the conjecture has been given in the form of two limit theorems, as n-. relating the random matrices to matrices formed from the Haar measure, h,, on the orthogonal group The present paper considers a reformulation of the conj...

For mixing Zd-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown that directional mixing and directional convergence of the uniform measure supported on periodic points to Haar measure occurs at a uniform rate ind...