The Gaussian free field and Hadamard's variational formula (Q2249582)

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.