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We study solutions of 12 (-i - A (x) ) ²f= f on domains R² with Dirichlet boundary conditions and prove exponential decay estimates in terms of an Agmon type distance to a classically allowed region. This metric depends only on the eigenvalue and associated magnetic field. In fact the main quantity in the weight for this distance function can be interpreted as an `average magnetic flux' of the magnetic field along all domains whose boundary is the closed curved formed by joing the `minimal path' to another random path. Our estimates are based on an analysis of the associated heat kernel using the Feynman-Kac-It\'o formula.
Hadrian Quan (Sun,) studied this question.