We describe the Schwinger–Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems.

我们描述了与自由差商相关的施温格-戴森方程。这样的方程出现在不同的领域，例如组合数学（通过平面映射的枚举问题）、算子代数（通过自由概率中各部分自然积分的定义）、经典概率（通过排斥矩阵中的随机矩阵或粒子）相互作用）。在这些讲义中，我们将讨论该方程何时唯一地定义系统，以及在这种情况下它如何导致解决方案的深层属性。该分析可以扩展到近似满足这些方程的系统，例如随机矩阵或库仑气体相互作用粒子系统。