Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Abstract

Abstract. We compute the bi-free max-convolution which is the operation on bi-variate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely-divisible laws their determination becomes in this way a classical analysis question. IntroductionThe definition and classification of free max-stable laws in [2] had been an unexpected addition to the list of free probability analogues to classical probability items. Here we take the first step in a similar direction in bi-free probability [9]. We show that there is a simple formula for computing the bi-free extremal convolution of probability measures in the plane. This corresponds to computing the distribution of (a ∨ c, b ∨ d), where (a, b) and (c, d) are two bi-free two-faced pairs of commuting hermitian operators. Like the free extremal convolution on R defined in [2], the bi-free extremal convolution in the plane reduces the question of bi-free max-stable laws in the plane to an analysis problem in the classical context which we won't pursue here.