Free diffusions and matrix models with strictly convex interaction (Q841086)

The authors study solutions to the free stochastic differential equation \[ d X_t = d S_t - \tfrac{1}{2} DV( X_t) \,dt, \quad t \in [0, \infty), \quad \text{with } X_0 = Z,\tag{1} \] where \(V\) is a locally convex polynomial potential in \(m\) non-commuting variables and \(S\) is an \(m\)-dimensional free Brownian motion. The main purpose of this paper is to study the tracial states \(\tau_V\) which solve the Schwinger-Dyson equation \[ \tau_V \otimes \tau_V ( \partial_i P) = \tau_V ( D_i V P) \tag{2} \] for some general potential \(V\), where \(P\) is a polynomial, \(i \in \{ 1, \dots, m \}\), and \(\partial_i\) (resp., \(D_i\)) are respectively the non-commuting derivative (resp., the cyclic derivative) with respect to the variable \(X_i\). When \(V\) \(\in\) \({\mathbb C} \langle X_1, \dots, X_m \rangle\) is a polynomial in \(m\) non-commuting variables, then \(V\) is said to be \((c, M)\)-convex if for any \(m\)-tuple of non-commutative variables \(X =\) \(( X_1, \dots, X_m )\) and \(Y\) \(=\) \(( Y_1, \dots, Y_m )\) in some \(C^*\)-algebra \(( {\mathcal A}, \| \cdot \|_{\infty} )\) satisfying \(\| X_i \|_{\infty}\), \(\| Y_i \|_{\infty} \leqslant M\) for \(i =\) \(1, \dots, m\), we have the inequality \[ [ D V(X) - D V(Y) ] \cdot ( X - Y) \geq c ( X- Y) \cdot (X - Y), \tag{3} \] where the inequality is understood in the sense of operators, i.e., \(X \geq Y\) iff \(X - Y\) is self-adjoint and has non-negative spectrum. Assume that \(V\) is \((c, M)\)-convex. Let \(( {\mathcal A}, *, \phi)\) be a non-commutative probability space generated by a free Brownian motion \(S\), and \(\| \cdot \|_{\infty}\) be the operator norm in \(( {\mathcal A}, \phi)\). Then there exists a unique solution \(X_t^Z\) to (1) such that \[ \| X_t^Z \|_{\infty} \leqslant M, \quad \forall t \in [0, \infty), \quad \text{and} \quad \limsup_{t \to \infty} \| X_t^Z \|_{\infty} \leqslant B_0, \tag{4} \] where \(B_0\) is a finite constant depending only on \(c\) and \(\| DV(0).DV(0) \|_{\infty}\). The authors prove that such free processes have a unique stationary law \(\mu_V\), and also that the restriction \(\tau_V\) of \(\mu_V\) to \({\mathbb C} \langle X_1, \dots, X_m \rangle\) satisfies the Schwinger-Dyson equation, namely, for all \(P \in\) \({\mathbb C} \langle X_1, \dots, X_m \rangle\) and all \(i \in\) \(\{ 1, \dots, m \}\), \(\tau_V\) satisfies \[ \tau_V \otimes \tau_V ( \partial_i P ) = \tau_V ( D_i V P). \tag{5} \] When the potential \(V\) is self-adjoint, the authors show that the law \(\mu_V\) is the limit law of a random matrix model, in which an \(m\)-tuple of self-adjoint matrices \(A_1, \dots, A_m\), are chosen according to the law \[ d \mu_V^N(A_1, \dots, A_m) = \frac{1}{Z_N(V)} \exp \{ - N \cdot \text{Tr}( V(A_1, \dots, A_m)) \} \,d A_1\dots d A_m. \tag{6} \] If \(V = V_{\beta}\) depends on complex parameters \(\beta_1, \dots, \beta_k\), it is shown that the moments of the law \(\mu_V\) are analytic in \(\beta=( \beta_i )_{1 \leqslant i \leqslant k}\) at least for those \(\beta\) for which \(V_{\beta}\) is locally convex. In particular, this gives information on the region of convergence of the generating function for the enumeration of related planar maps. Define \[ \begin{aligned} \hat{\mu}_N(P) & = \tfrac{1}{N} \,\text{Tr} ( P(A_1,\dots, A_m)), \\ d \mu_V^N ( A_1, \dots, A_m) & = \frac{1_{ \{ \| A_i \|_{\infty} \leqslant M \} }}{ Z_N(V)} e^{- N \cdot \text{Tr}( V(A_1, \dots, A_m ))} \,d A_1 \cdots d A_m, \end{aligned} \] and \[ \bar{\mu}_V^N(P) := \int \hat{\mu}_N (P) \,d \mu_V^N ( A_1, \dots, A_m ). \tag{7} \] Let \(\tau\) be the unique law of \(m\) self-adjoint variables such that for all \(i \in \{ 1, \dots, m \}\) and all polynomials \(P\), \[ \tau \otimes \tau( \partial_i P) = \tau( D_i V P) \tag{8} \] and such that \(\tau( X_i^{2k} ) \leqslant B_0^{2k}\), for any \((c, M)\)-convex self-adjoint polynomial \(V\). Then it is proved that \(\bar{\mu}_V^N\) converges to \(\tau\), and also that if \(Z\) has law \(\tau\), then the algebra \(W^*(Z)\) can be embedded into the ultrapower of the hyperfinite \(II_1\) factor. The microstate \(\Gamma_t\) is given by \[ \begin{aligned} \Gamma_t \equiv \Gamma( \tau, \varepsilon, R, k) = \Big\{&X_1, \dots, X_m : \| X_i \|_{\infty} \leqslant R,\;| \tfrac{1}{N} \text{Tr}( P(X_1, \dots, X_m )) - \tau(P) | \leqslant \varepsilon, \\ &\text{for all monomials of degree less than }k \Big\}. \end{aligned}\tag{9} \] Let vol denote the volume on the space of \(m\) \(N \times N\) Hermitian matrices. Then the free entropy \(\chi (\tau)\) satisfies \[ \begin{aligned} \chi(\tau) & = \limsup _{\substack{ \varepsilon \downarrow 0 \\ k, R \to \infty }} \limsup_{N \to \infty} \left( \frac{1}{N^2} \log \text{vol} ( \Gamma_t) + \frac{m}{2} \log N \right) \\ & = \liminf _{\substack{ \varepsilon \downarrow 0 \\ k, R \to \infty }} \liminf_{N \to \infty} \left( \frac{1}{N^2} \log \text{vol} ( \Gamma_t) + \frac{m}{2} \log N \right). \end{aligned} \tag{10} \] Since the solution \(X_t\) has nice convergence properties with respect to the operator norm as \(t\) goes to infinity, reflecting on various properties of \(C^*\) and \(W^*\) algebras generated by an \(m\)-tuple with law \(\mu_V\), the authors can show that the microstates free entropy \(\chi(\mu_V)\) is finite when \(V\) is self-adjoint. Moreover, the fact that the support of the law of any self-adjoint polynomial in \(X_1, \dots, X_n\) under the law \(\mu_V\) is connected, is derived as well. For other related works, see, e.g., \textit{P.\,Biane} [Acta Math.\ Sin., Engl.\ Ser.\ 19, No.\,3, 497--506 (2003; Zbl 1040.46042)] for matrix models and free entropy, \textit{P.\,Biane} and \textit{R.\,Speicher} [Ann.\ Inst.\ Henri Poincaré, Probab.\ Stat.\ 37, No.\,5, 581--606 (2001; Zbl 1020.46018)] for free diffusions and free entropy, and \textit{U.\,Haagerup, H.\,Schultz} and \textit{S.\,Thorbjørnsen} [Adv.\ Math.\ 204, No.\,1, 1--83 (2006; Zbl 1109.15020)] for a random matrix approach to the lack of projections in the algebra.