Euclidean Field Theories in 3D: Nonlinear Wave Equations and Phase Transitions

Abstract

In this thesis we are interested in the statistical mechanics of Euclidean field theories in 3D. We solve two problems: the first concerns the relationship between Gaussian measures and nonlinear wave equations; the second concerns phase transitions for $\phi^4_3$. The common theme between our contributions is the development of the variational approach of Barashkov and Gubinelli [BG19] to ultraviolet stability, which allows one to control the singular short-distance behaviour of Euclidean field theories in 3D, in the context of statistical mechanics arguments.

Our first contribution is to establish the quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the cubic defocusing wave equation. This extends previous work in the two-dimensional case [OT20]. Two new ingredients in the three-dimensional case are (i) the construction of certain weighted Gaussian measures based on the variational approach to ultraviolet stability, and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions. This is joint work with Tadahiro Oh, Nikolay Tzvetkov, and Hendrik Weber [GOTW18].

Our second contribution is to quantify the phase transition for $\phi^4_3$. In particular,
we establish a surface order large deviation estimate for the magnetisation of low temperature $\phi^4_3$. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $\phi^4_3$ singular stochastic PDE. Our main technical results are contour bounds for $\phi^4_3$, which extends 2D results by Glimm, Jaffe, and Spencer [GJS75]. We adapt an argument by Bodineau, Velenik, and Ioffe [BIV00] to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of $\phi^4_3$ whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $\phi^4_3$. This is joint work with Ajay Chandra and Hendrik Weber [CGW20].

Date of Award	2 Dec 2020
Original language	English
Awarding Institution	University of Bath
Supervisor	Hendrik Weber (Supervisor) & Marcel Ortgiese (Supervisor)