Matrix Whittaker processes

Area of Science:

• Probability Theory
• Stochastic Processes
• Matrix Analysis

Background:

• The study builds upon existing models of random walks and polymer partition functions.
• It addresses the need for generalized models in higher dimensions.

Purpose of the Study:

• To introduce and analyze a discrete-time Markov process on triangular arrays of matrices.
• To establish conditions for autonomous Markovian evolution of the process's bottom edge.
• To define and characterize a matrix Whittaker measure.

Main Methods:

• Utilizing inverse Wishart random matrices to drive the process.
• Establishing intertwining relations to prove Markovian evolution.
• Employing Laplace approximation and constrained minimization for analysis.

Main Results:

• Demonstrated autonomous Markovian evolution for the bottom edge under specific initial configurations.
• Defined and identified a matrix Whittaker measure for a singular initial configuration.
• Connected the process to a d-dimensional generalization of log-gamma polymer partition functions.

Conclusions:

• The research provides a new framework for studying matrix-valued stochastic processes.
• The findings offer insights into the behavior of random matrix models and their applications.