The partition function of the Symmetric Matrix Ensemble is identified with the \(\tau \)-function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the \(\tau \)-function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

中文翻译：

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对称矩阵系综和可积流体动力链

对称矩阵系综的配分函数由Pfaff 格的特定解的\(\tau \)函数标识。我们表明，在偶数功率相互作用的情况下，在热力学极限中，\(\tau \)函数对应于流体力学类型的可积链的解。我们证明如此获得的流体动力学链是可对角化的，并且允许任意数量的组件中黎曼不变量的流体动力学减少。