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/

Zendy Plus

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

Zendy Tools

Zendy Open

Motivated by the asymptotic collective behavior of random and deterministicmatrices, we propose an approximation (called "free deterministic equivalent")to quite general random matrix models, by replacing the matrices with operatorssatisfying certain freeness relations. We comment on the relation between ourfree deterministic equivalent and deterministic equivalents considered in theengineering literature. We do not only consider the case of square matrices,but also show how rectangular matrices can be treated. Furthermore, weemphasize how operator-valued free probability techniques can be used to solveour free deterministic equivalents.  As an illustration of our methods we consider a random matrix model studiedfirst by R. Couillet, J. Hoydis, and M. Debbah. We show how its freedeterministic equivalent can be treated and we thus recover in a conceptual waytheir result.  On a technical level, we generalize a result from scalar valued freeprobability, by showing that randomly rotated deterministic matrices ofdifferent sizes are asymptotically free from deterministic rectangularmatrices, with amalgamation over a certain algebra of projections.  In the Appendix, we show how estimates for differences between Cauchytransforms can be extended from a neighborhood of infinity to a region close tothe real axis. This is of some relevance if one wants to compare the originalrandom matrix problem with its free deterministic equivalent.

Free Deterministic Equivalents, Rectangular Random Matrix Models, and Operator-Valued Free Probability Theory