2017/03/31 by Makoto Katori
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Affine transformation #Dimension (graph theory) #Kernel (algebra) #Mathematics #Pure mathematics #Random Matrices and Applications #Stochastic processes and statistical mechanics #cond-mat.stat-mech #math-ph #math.MP #math.PR #msc:17B22 #msc:33E05 #msc:60B20 #msc:60G44 #msc:60J65 #msc:82C22 #nlin.SI
paper · pdf · doi:10.3842/sigma.2017.079
published as SIGMA 13 (2017), 079, 36 pages
arxiv created 2017/10/04 · openalex publication_date 2017/10/04 · arxiv updated 2017/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce seven families of stochastic systems of interacting particles in onedimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families A N -1 , B N , C N and D N , we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.