Published online by Cambridge University Press: 20 November 2025
Complementing the presentation of the Gaussian free field (GFF) with zero boundary conditions in Chapter 1, and on manifolds in Chapter 5, we devote this chapter to studying further variants of the field. The main example is the GFF in two dimensions with Neumann (or free) boundary conditions. We give a rigorous definition of this field as both a stochastic process indexed by suitable test functions and as a random distribution modulo constants. As in Chapter 1, we show the equivalence of these two viewpoints; however, in this case, further analytical arguments are required. We describe the covariance function of the field, prove key properties such as conformal covariance and the spatial Markov property, and discuss its associated Gaussian multiplicative chaos measures on the boundary of the domain where it is defined. We also cover the definition and properties of the whole plane Gaussian free field and the Gaussian free field with Dirichlet–Neumann boundary conditions, building on the construction of the Neumann GFF. We further prove that the whole plane GFF can be decomposed into a Dirichlet part and a Neumann part. Finally, we show that the total mass of the GMC associated to a Neumann GFF on the unit disc is almost surely finite.
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