Free diffusions and Matrix models with strictly convex interaction
| dc.creator | Guionnet, A. | |
| dc.creator | Shlyakhtenko, D. | |
| dc.date | 2007-01-26 | |
| dc.date.accessioned | 2026-07-07T07:43:34Z | |
| dc.date.available | 2026-07-07T07:43:34Z | |
| dc.description | We study solutions to the free stochastic differential equation $dX_t = dS_t - \half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$ non-commuting variables. We show that for self-adjoint $V$, the law $μ_V$ 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 $\exp(-N \textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_β$ depends on complex parameters $β_1,...,β_k$, then the law $μ_V$ is analytic in $β$ at least for those $β$ for which $V_β$ 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 $dX_t$ 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 $μ_V$. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free entropy $χ(τ_V)$ is finite. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $X_1,...,X_n$ under the law $μ_V$ is connected, vastly generalizing the case of a single random matrix. | |
| dc.identifier | https://arxiv.org/abs/math/0701787 | |
| dc.identifier | http://arxiv.org/abs/math/0701787 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/122878 | |
| dc.subject | Operator Algebras | |
| dc.subject | Probability | |
| dc.subject | 46L54 (Primary), 60H99, 05C30 (Secondary) | |
| dc.title | Free diffusions and Matrix models with strictly convex interaction | |
| dc.type | text |