2007/01/26 by Alice Guionnet, Guionnet, A., Dimitri Shlyakhtenko +1 · 2 citations
Mathematics · #05C30 (Secondary) #46L54 (Primary) #60H99 #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/0701787
openalex publication_date 2007/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study solutions to the free stochastic differential equation dXt = dSt - \half DV(Xt)dt, where V is a locally convex polynomial potential in m non-commuting variables. We show that for self-adjoint V, the law μV of a stationary solution is the limit law of a random matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law exp(-N \textrmTr(V(A1,...,Am)))dA1... dAm. We show that if V=Vβ depends on complex parameters β1,...,βk, then the law μV is analytic in β at least for those β for which Vβ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution dXt has nice convergence properties with respect to the operator norm. This allows us to derive several properties of C^* and W^* algebras generated by an m-tuple with law μV. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II1 factor. We show that the microstates free entropy χ(τV) is finite. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in X1,...,Xn under the law μV is connected, vastly generalizing the case of a single random matrix.