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تسجيل مستخدم جديدAn Elementary Approach to Free Entropy Theory for Convex Potentials
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تأليف
David Jekel
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We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDEs, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(mathbb{C})_{sa}^m$ to prove the following. Suppose $mu_N$ is a probability measure on on $M_N(mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $mu_N$ converge to a non-commutative law $lambda$. Moreover, the free entropies $chi(lambda)$, $underline{chi}(lambda)$, and $chi^*(lambda)$ agree and equal the limit of the normalized classical entropies of $mu_N$.
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