Abstract
In this paper we study certain edge-weighted random walks on infinite graphs with bounded vertex degree for which the second smallest eigenvalue of the Laplacian is negative.We find analogues of the Feynman-Kac functional and give several conditions equivalent to the boundeness of the corresponding gauge function.From this result we derive a new formula for the second smallest eigenvalue of the Laplacian and we apply this formula to the theory of graph coverings.An analogue of the conditional gauge theorem is shown to hold for certain Schrodinger operators.
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