For a Dirichlet form $(\mathcal {E},\mathcal {F})$ on L2(E;m), let $\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}$ be the Gaussian field indexed by the extended Dirichlet space $\mathcal {F}_{e}$. We first solve the equilibrium problem for a regular recurrent Dirichlet form $\mathcal {E}$ of finding for a closed set B a probability measure μB concentrated on B whose recurrent potential $R\mu ^{B}\in \mathcal {F}_{e}$ is constant q.e. on B (called a Robin constant). We next assume that E is the complex plane $\mathbb {C}$ and $\mathcal {E}$ is a regular recurrent strongly local Dirichlet form. For the closed disk $\bar B(\textbf {x},r)=\{\textbf {z}\in \mathbb {C}:|\textbf {z}-\textbf {x}|\le r\}$, let μx, r and f(x, r) be its equilibrium measure and Robin constant. Denote the Gaussian random variable $X_{R\mu ^{\textbf {x}.r}}\in \mathbb {G}(\mathcal {E})$ by Yx, r and let, for a given constant γ > 0, $\mu _{r}(A,\omega )={\int \limits }_{A} \exp (\gamma Y^{\textbf {x},r}-(1/2)\gamma ^{2} f(\textbf {x},r))d\textbf {x}.$ Under a certain condition on the growth rate of f(x, r), we prove the convergence in probability of μr(A, ω) to a random measure $\overline {\mu }(A,\omega )$ as r↓ 0. The possible range of γ to admit a non-trivial limit will then be examined in the cases that $(\mathcal {E}.\mathcal {F})$ equals $(\frac 12{\textbf {D}}_{\mathbb {C}},H^{1}(\mathbb {C}))$ and $(\textbf {a},H^{1}(\mathbb {C}))$, where a corresponds to the uniformly elliptic partial differential operator of divergence form.