Radial $Φ^4$ models in supercritical regimes

Abstract

We consider the $Φ^4_d$ measure under radial symmetry on the unit ball, formally given by $$dρ(u) = \frac 1 Z \exp\Big( - \frac 14 \int_{B_1} |u|^4 dx - \frac 12 \int_{B_1} |\nabla u|^2 dx\Big) d u, $$ where $u$ is restricted to radially symmetric functions with 0 boundary conditions. We prove that for every dimension $d$, there exists such a measure, which is concentrated on $L^p(B_1)$ for every $p < \frac{2d}{d-2}$ (for every $p < \infty$ if $d \le 2$). Furthermore, we show that these measures are not Gaussian, and when $d \ge 4$, they are singular with respect to the underlying Gaussian measure. We achieve this result by first defining the measure $ρ^+_δ[λ]$ on the shell $\{δ\le |x| \le 1\}$, $0 < δ< 1$, with a boundary condition on the sphere of radius $δ$ of the form $u|_{\{|x| = δ\}} = λ$. Then we prove that this measure satisfies a variational formula, and use it to show moment bounds which are uniform in the boundary value $λ$. A crucial ingredient in this argument is the fact that every solution of the elliptic equation $-Δu + u^3 = 0$ on the annulus $\{ δ\le |x| \le 1 \}$ satisfies uniform bounds inside the domain, independently of the boundary conditions.