Free Diffusions and Matrix Models with Strictly Convex Interaction
Free Diffusions and Matrix Models with Strictly Convex Interaction
复制标题
具有严格凸相互作用的自由扩散和矩阵模型
DOI：
10.1007/s00039-009-0704-0
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发表时间：
2007
影响因子：
2.2
通讯作者：
D. Shlyakhtenko
中科院分区：
文献类型：
--
作者：
A. Guionnet;D. Shlyakhtenko
研究自由随机微分方程的解，其中V是m个非交换变量和m维自由布朗运动中的局部凸多项式势。我们证明了这样的自由过程有唯一的平稳分布μV。当势维斯是自伴矩阵时，我们证明了μ维斯律是随机矩阵模型的极限律，其中m个自伴矩阵元组按exp$$选取（-NTr（V（A_{1}，\ldots，A_{m}）dA_{1} \cdots dA_{m}/ZN（V）$$.如果V =Vβ依赖于复参数，证明了μ V的矩至少对V β局部凸的那些β是在β中解析的.特别地，这给出了有关平面映射计数的生成函数的收敛区域的信息，我们证明了解X具有关于算子范数astgoes到无穷大的良好收敛性质。这使我们能够证明由具有律μ V的m-元组生成的C * 和W * 代数与由半圆系统生成的C * 和W * 代数具有许多相同的性质。其中之一是缺乏投影，精确性，Haagerup属性和嵌入超有限II 1因子的超幂。我们证明了当维斯自伴时微观态自由熵χ（μV）是有限的，这些结果的一个推论是：在μ维斯定律下，任何自伴多项式的定律的支集是连通的，我们还从这个动力学方法中推出了Haagerup和Thorbjornsen [HT]证明的GUE中独立矩阵的算子范数的收敛性。推广到矩阵通过凸势相互作用的情形。
We study solutions to the free stochastic differential equation, whereVis a locally convex polynomial potential inmnon-commuting variables andSanm-dimensional free Brownian motion. We prove that such free processes have a unique stationary distribution μV. When the potentialVis self-adjoint, we show that the law μVis the limit law of a random matrix model, in which anm-tuple of self-adjoint matrices are chosen according to the law exp$$(-NTr(V(A_{1},\ldots,A_{m})))dA_{1} \cdots dA_{m}/ZN(V)$$.IfV=Vβdepends on complex parameters, we prove that the moments of the law μVare analytic in β at least for those β for whichVβis locally convex. In particular, this gives information on the region of convergence of the generating function for the enumeration of related planar maps.We prove that the solutionXthas nice convergence properties with respect to the operator norm astgoes to infinity. This allows us to show that theC*andW*algebras generated by anm-tuple with law μVshare many properties with those generated by a semi-circular system. Among them is the lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II1factor. We show that the microstates free entropy χ(μV) is finite whenVis self-adjoint.A corollary of these results is the fact that the support of the law of any self-adjoint polynomial inunder the law μVis connected, vastly generalizing the case of a single random matrix.We also deduce from this dynamical approach that the convergence of the operator norms of independent matrices from the GUE proved by Haagerup and Thorbjornsen [HT] extends to the context of matrices interactingviaa convex potential.