English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce real second-order freeness in second-order noncommutativeprobability spaces. We demonstrate that under this definition, three realmodels of random matrices, namely real Ginibre matrices, Gaussian orthogonalmatrices, and real Wishart matrices, are asymptotically second-order free.These ensembles do not satisfy the complex definition of second-order freenesssatisfied by their complex analogues. We use a combinatorial approach to thematrix calculations similar to the genus expansion for complex random matrices,but in which nonorientable surfaces appear, demonstrating the commonalitybetween the real models and the distinction from their complex analogues,motivating this distinct definition. In the real case we find, in addition tothe terms appearing in the complex case corresponding to annular spokediagrams, an extra set of terms corresponding to annular spoke diagrams inwhich the two circles of the annulus are oppositely oriented, and in which thematrix transpose appears.

Real second-order freeness and the asymptotic real second-order freeness of several real matrix ensembles