Phase transitions for $\phi^4_3$

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 contributions are contour bounds for $\phi^4_3$, which extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by Bodineau, Velenik, and Ioffe (2000) 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 $\phi^4_3$ developed recently by Barashkov and Gubinelli (2019).