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Register a new userAn Elementary Approach to Free Entropy Theory for Convex Potentials
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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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Let $(X_1,dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $mathrm{W}^*(X_1,dots,X_m) to mathrm{W}^*(S_1,dots,S_m)$ which maps $mathrm{W}^*(X_1,dots,X_k)$ to $mathrm{W}^*(S_1,dots,S_k)$ for each $k = 1, dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,dots,X_m)$ relative to the law of $(S_1,dots,S_m)$.
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