Phase Transitions for $$\phi ^4_3$$

Abstract

AbstractWe establish a surface order large deviation estimate for the magnetisation of low temperature $$\phi ^4_3$$ ϕ 3 4 . As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $$\phi ^4_3$$ ϕ 3 4 singular stochastic PDE. Our main technical contributions are contour bounds for $$\phi ^4_3$$ ϕ 3 4 , which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 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 $$\phi ^4_3$$ ϕ 3 4 whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $$\phi ^4_3$$ ϕ 3 4 developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020).