Random Gaussian fields applied to fluctuating polymers and biopolymers, eg protein chains and DNA. First, non-conducting knotted charged strings or polymers in a non-conducting viscous fluid leads to the concept of knot energy functionals. The electrosatic energy will minimise without intersection due to Coulomb forces. One can then define a knot-energy funcional. Let $(K,\bar{K})$ be two such knots in a domain $\mathbb{D}$ such that $(K,\bar{K})\subset\mathbb{D}\subset\mathbb{R}^{3}$. Let $x^{i}\subset K$ and $y^{j}\subset \bar{K}$. A regulated knot energy functional $\mathbf{\mathcal{K}ef}(K,\bar{K})$ is defined as

$\mathbf{\mathcal{K}ef}(K,\bar{K}) = - \delta^{ij}lim_{\epsilon \rightarrow 0} \bigg({\displaystyle \int_{K}}{\displaystyle \int_{\bar{K}}}$ ${\displaystyle \frac{dx^{i} dy^{j} } {|x-y|^{2}}}-\frac{2}{\epsilon}\bigg)$

Similary, two knots carrying unit currents leads to the Bio-Savart law

$\mathbf{\mathcal{K}f}(K,\bar{K}) = \delta_{ij}\bigg({\displaystyle \int_{K}}{\displaystyle \int_{\bar{K}}}$ ${\displaystyle \frac{dx^{i} dy^{j} } {|x-y|} }\bigg)$

$\bf{Theorem}$

Let $\widehat{\mathcal{Q}}_{i}(x,\omega)$ be a random Gaussian vector field defined for all $x\in\mathbb{D}\subset\mathbb{R}^{3}$ and with respect to a triplet $\Omega,\mathcal{F},\mathbf{P})$. Then if:

$\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)={\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega) =0$

$\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)\otimes \widehat{\mathcal{Q}}_{j}(y,\omega)$

$={\displaystyle\int}_{\Omega}d\mathbf{P}{\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega)(\omega)\widehat{\mathcal{Q}}_{j}(y,\eta))= {\displaystyle \frac{\zeta^{2}}{|x-y|^{2}}}$

The expectation of the stochastic Wilson line of $\mathcal{Q}_{i}(x,\omega)$ gives

$\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg({\displaystyle\int_{K}} \widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)$$exp \bigg({\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta^{2} \frac{dx^{i}dy^{j}}{|x-y|^{2}}}\bigg)$

$\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp \bigg(i{\displaystyle\int_{K}} \widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)$$exp \bigg(-{\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta^{2} \frac{dx^{i}dy^{j}}{|x-y|^{2}}}\bigg)$

if:

$\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)={\displaystyle \int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega) =0$

$\mathbf{E}(\widehat{\mathcal{Q}}_{i}(x,\omega)\otimes \widehat{\mathcal{Q}}_{j}(y,\omega)$

$={\displaystyle\int}_{\Omega}d\mathbf{P}{\displaystyle\int}_{\Omega}d\mathbf{P}(\omega)\widehat{\mathcal{Q}}_{i}(x,\omega)(\omega)\widehat{\mathcal{Q}}_{j}(y,\eta))= {\displaystyle \frac{\zeta}{|x-y|}}$

The expectation of the stochastic Wilson line of $\widehat{\mathcal{Q}}_{i}(x,\omega)$ gives

$\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg({\displaystyle\int_{K}}\widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)$$exp \bigg({\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta \frac{dx^{i}dy^{j}}{|x-y|}}\bigg)$

$\mathbf{E}(\widehat{W}[K])={\displaystyle \int_{\Omega}}d\mathbf{P}(\omega)exp\bigg(i{\displaystyle\int_{K}}\widehat{\mathcal{Q}}_{i}(x,\omega)dx^{i}\bigg)$$exp \bigg(-{\displaystyle \int_{K}\int_{\bar{K}}\delta_{ij}\zeta \frac{dx^{i}dy^{j}}{|x-y|}}\bigg)$