论文标题
矩阵惠特克过程
Matrix Whittaker processes
论文作者
论文摘要
我们研究了由倒数随机矩阵驱动的大小$ d \ geq 1 $矩阵矩阵三角阵列的离散时间过程。右边缘的组成部分随着带有单方面相互作用的正定义矩阵的乘法随机行动而演变，并且可以看作是log-gamma聚合物分区函数的$ d $维概括。我们建立了交织的关系，以证明，对于三角形过程的适当初始配置，底部边缘具有具有显式过渡内核的自主马尔可夫进化。然后，我们表明，对于特殊的单数初始配置，底部边缘的固定时间定律是矩阵惠特克度量，我们定义了它。为了实现这一目标，我们执行了一个拉普拉斯近似，该近似需要解决矩阵参数在有向图上的某些能量函数的约束最小化问题。
We study a discrete-time Markov process on triangular arrays of matrices of size $d\geq 1$, driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a $d$-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs.
