The random variable 1 + z ₁ + z ₁ z ₂ + … appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to N N matrices either real symmetric β = 1 or complex Hermitian β = 2. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to N fermions in a Morse potential, which are non-interacting for β = 2. At finite N the distribution of eigenvalues exhibits heavy tails, generalizing Kesten’s results in the scalar case. The density of fermions in this potential is studied for large N, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large N limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valk, Grabsch and Texier, as well as Ossipov, is discussed.

随机变量 1 + z ₁ + z ₁ z ₂ + ... 出现在许多上下文中，并被 Kesten 证明表现出重尾分布。我们考虑这个变量的自然扩展及其对N N 个矩阵的关联递归，要么是实对称β = 1 要么是复厄米特β = 2。在这个递归的连续极限中，我们表明矩阵分布收敛于随机矩阵。使用映射到摩尔斯势中的N 个费米子来解决完整的动力学问题，对于β = 2 ，这些费米子是非相互作用的。 在有限的N 处特征值的分布表现出重尾，在标量情况下概括了 Kesten 的结果。对于大N研究了该势中费米子的密度，特征值分布的幂律尾与所谓的行列式贝塞尔过程的性质有关，该过程描述了随机矩阵的硬边普遍性。对于离散矩阵递归，使用大N限制中的自由概率，我们获得了平稳分布的自洽方程。讨论了我们的结果与 Rider 和 Valk、Grabsch 和 Texier 以及 Ossipov 近期作品的关系。