Central Measures of Continuous Graded Graphs:\\ the Case of Distinct Frequencies

TL;DR

This paper introduces a class of continuous graded graphs, characterizes their ergodic central measures, and links these measures to spectral descriptions of infinite-dimensional matrix ensembles, offering new insights into their structure.

Contribution

It provides a novel framework for understanding ergodic central measures on continuous graded graphs and connects these measures to spectral measures of infinite matrices.

Findings

Characterization of ergodic central measures on continuous graded graphs.

Connection between central measures and spectral measures of infinite matrices.

New spectral description of infinite-dimensional Wishart measures.

Abstract

We define a class of continuous graded graphs similar to the graph of Gelfand--Tsetlin patterns, and describe the set of all ergodic central measures of discrete type on the path spaces of such graphs. The main observation is that an ergodic central measure on a subgraph of a Pascal-type graph can often be obtained as the restriction of the standard Bernoulli measure to the path space of the subgraph. This observation dramatically changes the approach to finding central measures also on discrete graphs, such as the famous Young graph. The simplest example of this type is given by the theorem on the weak limits of normalized Lebesgue measures on simplices; these are the so-called Ces\`aro measures, which are concentrated on the sequences with prescribed Ces\`aro limits (this limit parametrizes the corresponding measure). More complicated examples are the graphs of continuous Young diagrams with fixed number of rows and the graphs of spectra of infinite Hermitian matrices of finite rank. We prove existence and uniqueness theorems for ergodic central measures and describe their structure. In particular, our results 1) give a new spectral description of the so-called infinite-dimensional Wishart measures~\cite{W}~ -- ergodic unitarily invariant measures of discrete type on the set of infinite Hermitian matrices; 2) describe the structure of continuous analogs of measures on discrete graded graphs. New problems and connections which appear are to be considered in new publications.