标题： 随机矩阵系综与可积微分恒等式 显示英文标题

标题： Random matrix ensembles and integrable differential identities

摘要： 可积微分恒等式，连同与系相关的初值条件，为随机矩阵理论中相关可观测量和状态函数的刻画提供了一种有效方法。 我们针对酉系和正交系发展了这种方法。 特别是，我们将注意力集中在概率测度由一个表示为偶相互作用项形式级数的哈密顿量诱导的约化上。 我们证明，与Volterra格子相关的酉系序参量提供了修正KP方程的一个解。 对于与Pfaff格子相关的正交系的类似约化，得到了一个新的可积链。 计算正交系序参量的关键步骤是利用从正交多项式到斜正交多项式的映射来求初始条件。 热力学极限导致了一个具有流体力学类型的可积系统（正交系中的链）。 有趣的是，我们发现离散系统的初值问题及其连续极限的解均由同一个半离散动力学链给出。 显示英文摘要

摘要： Integrable differential identities, together with ensemble-specific initial conditions, provide an effective approach for the characterisation of relevant observables and state functions in random matrix theory. We develop this approach for the unitary and orthogonal ensembles. In particular, we focus on a reduction where the probability measure is induced by a Hamiltonian expressed as a formal series of even interaction terms. We show that the order parameters for the unitary ensemble, that is associated with the Volterra lattice, provide a solution of the modified KP equation. The analogous reduction for the orthogonal ensemble, associated with the Pfaff lattice, leads to a new integrable chain. A key step for the calculation of order parameters for the orthogonal ensemble is the evaluation of the initial condition by using a map from orthogonal to skew-orthogonal polynomials. The thermodynamic limit leads to an integrable system (a chain for the orthogonal ensemble)of hydrodynamic type. Intriguingly, we find that the solution to the initial value problem for both the discrete system and its continuum limit are given by the very same semi-discrete dynamical chain.