Abstract
We now consider a possibly site-dependent “magnetic field”, i.e. a vector \(h = {({h_x})_{x \in L}}\) . In the presence of a magnetic field the random-walk expansion of S n (x 1,...,x n ) undergoes a double alteration. On the one hand we must also include paths connecting one x i with an “external magnetic field”. This means that the walks may end up not only at the points x 1,...,x n (“lattice sources”), but also at any other point in the lattice, where they may be thought to reach an “h-source”. This is nothing more than the familiar combinatorics of the Gaussian model (free field) in nonzero magnetic field. On the other hand the expression for the weight depends on the kind of source each path has as endpoint, and it may happen that the same walk must be counted twice with different weights. We are forced to complicate the notation to distinguish the two cases.
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© 1992 Springer-Verlag Berlin Heidelberg
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Fernández, R., Fröhlich, J., Sokal, A.D. (1992). Random-walk models in the presence of a magnetic field. In: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02866-7_10
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DOI: https://doi.org/10.1007/978-3-662-02866-7_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-02868-1
Online ISBN: 978-3-662-02866-7
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