We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan’s isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the β-Dyson’s Brownian motion. For this is a consequence of the Gaussian case, however the relation holds for general β. We further raise the question whether there is an analogue of β-Dyson’s Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case we give a simple construction.
This work was supported by the French National Research Agency (ANR) grant within the project MALIN (ANR-16-CE93-0003).
The author thanks Guillaume Chapuy and Jérémie Bouttier for discussions and references on the beta ensembles. The author thanks Yves Le Jan and Wendelin Werner for their feedback on the preliminary version of the article.
"Isomorphisms of β-Dyson’s Brownian motion with Brownian local time." Electron. J. Probab. 26 1 - 31, 2021. https://doi.org/10.1214/21-EJP697