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The Gaussian free field and Hadamard’s variational formula. (English) Zbl 1302.60134

The authors relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator – called the Hadamard operator – associated with a given flow of growing domains. The Hadamard operator is obtained by integrating a Poisson kernel, the normal derivative, of the Green function.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
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