Séminaire de Probabilités

The asymptotic limit of random tensor flattenings

by Ion Nechita

Amphi Schwartz

Amphi Schwartz

Given a random tensor with i.i.d. entries, we consider the collection of all its flattenings (reshaping into matrices). We study the joint limit of these matrices in the sense of free probability. Under mild assumptions, we determine their limiting ∗-distribution in simple terms. We show that the limit is an operator-valued circular family, over the algebra CS_k of permutations. We identify free subfamilies and present applications to quantum information theory. This is joint work with Stéphane Dartois and Camille Male.