1- is just a dirac approximation with a Cauchy function. A good reference of the Combe-Thomas estimate is the book Random Operators: Disorder Effects on Quantum Spectra and Dynamics . Indeed if you just want symmetry it is should be much simpler than the semi-circle law.

Perfect! That's exactly what I need. (and thanks Kesten again by the way.)

This is a great reference. Thanks! But is it really enough ? I wish one could get $Q(X,t)$ in the numerator ?

With the new assumption of low bounded for A, It seems to me that you want is a kind of Harnack's inequality for $(W_t,M_t)$. (so that the probability density of is lower bounded.)

As you explain : $F=dA$ and magnectic monopole are inconsistent. Therefore I am not sure about what you want. (Still one can write $*d*F=J_e$ and $dF=J_m$.)