- y|]{} for $x,y$ with $|x-y|\ge L^{\frac{1+\rho}{\alpha}} = L^{7/8}$.

In a more traditional MSA approach, the value of the exponent $m$ (often referred to as the “mass”) depends upon the scale $L_k \sim (L_0)^{\alpha^k}$, $\alpha>1$, and the transition from $L_k$ to $L_{k+1}$ results in a tiny decay of the mass: $m_{k+1} = m_k(1 - o(1))$. Using the exponent $\gamma(m,L_k)$ allows to avoid rescaling the parameter $m$ itself and makes explicit the fact that the effective decay exponent is bounded from below by $m>0$, since $\gamma(m,L)> m$.

\[lem:cond.SubH.IID\] Consider a ball $\ball_L(u)$ and an operator $H=H_{\ball_L(u)}$ with fixed (non-random) potential $V$. Let $\{\psi_j, \, j=1, \ldots,  |\ball_L(u)|\}$ be the normalized eigenfunctions of $H$. Pick a pair of points $x',y' \in \ball_L(u)$ with $d(x',y') > 2(\ell + 1)$ and an integer $R \in[\ell+2, \, d(x',y') - (\ell+2)]$. Suppose that any ball $\ball_{\ell}(v)$ with $v\in\ball_R(x')$ is $(E,m)$-NS, and set \[eq:lem:ConSubH.IID\] q = q(, m) = e\^[-(m,)]{} Then:

1.  the kernel $\Pi_{\psi_j}(x,y)$ of the spectral projection $\Pi_{\psi_j} = \ketbra{\psi_j}{\psi_j}$ is $(\ell,q)$-subharmonic in $x \in \ball_R(x')$, with global maximum $\le 1$;

2.  if $\ball_L(u)$ is also $E$-NR, then the Green functions $G(x,y;E)=(H-E)^{-1}(x,y)$ are $(\ell,q)$-subharmonic in $x \in \ball_R(x')$, with global maximum $\le e^{L^\beta}$.

Localization and 