Exponential decay of fractional moments in the $H^{2|2}$ model on $\mathbb Z^2$

Source abstract

For the $H^{2|2}$ model on $\mathbb Z^2$, we prove that, at energy zero, fractional moment of Green functions, $\mathbb E[ G(x,y)^s]$ decays exponentially in the distance between $x$ and $y$, for every constant edge weight $W>0$ and every $0<s<1/2$. Here $G$ is the infinite-volume Green function at energy zero of the associated random Schrödinger operator. The main ingredient is a quantitative lower bound on the derivative of $\mathbb E[e^{u_x/2}]$ with respect to the edge weights, where $u$ is the pinned effective field.