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Matrix Whittaker processes.

Jonas Arista1, Elia Bisi2, Neil O'Connell3

  • 1Universität Bielefeld, Fakultät für Mathematik, Universitätsstrasse 25, 33615 Bielefeld, Germany.

Probability Theory and Related Fields
|September 1, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a novel discrete-time Markov process involving matrix-valued random walks. Researchers found that specific initial conditions lead to an autonomous evolution of the bottom edge, resulting in a matrix Whittaker measure.

Keywords:
Constrained minimisationDirected graphsInteracting Markov dynamicsIntertwining relationsNoncommutative polymer modelsWhittaker functions of matrix arguments

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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.