We study the ground state of the renormalized and non-renormalized Nelson model in quantum field theory. Localizations of the ground state are discussed. The ground state can be seen as a function of the position variable $x$ and the field variable $\phi$. We show the spatial exponential decay in $ x$ in terms of the Agmon metric and Gaussian domination in $\phi$ for the ground state. The super exponential decay of the number of bosons in the ground state is also shown. To show the localizations we apply the Gibbs measure associated with the ground state.
This talk is part of series of talks affiliated with the virtual Mittag-Leffler workshop "Scattering, microlocal analysis and renormalization", organized by Claudio Dappiaggi, Jacob Schach MÃ¸ller and Michal Wrochna. The full schedule can be found at: