Including the Haar measure we show that the effective potential of the regularized SU(2) Yang-Mills theory has a minimum at vanishing Wilsonline W = 0 for strong coupling, whereas it develops two degenerate minima close to W = ±1 for weak coupling. This suggests that the non-abelian character of SU(2) as contained in the Haar measure might be responsible for confinement. PACS number(s): 11.15.T...

We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are not Haar null. (Dougherty, answering an old question of Christensen, showed earlier that this was the case for some Polish, abelian, non-local...

Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure μ on G such that μ(B+g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G...

It is shown that for every closed, convex and nowhere dense subset C of a superreflexive Banach space X there exists a Radon probability measure μ on X so that μ(C + x) = 0 for all x ∈ X. In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a clos...

On logρ(AX)dX ≥ Cn log ‖A‖, where ‖A‖ denotes the operator norm of A (which equals the largest singular value of A), ρ denotes the spectral radius, and the integral is with respect to the Haar measure on On. The same result (with essentially the same proof) holds for the unitary group Un in place of the orthogonal group. The result does not hold in dimension 2.This answers questions asked in [3...

Abstract We study the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un,Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, LAn converges towards a rotationally invariant measure μ on the complex plane...