Brownian loop soup (${\mathcal{L}}_{\lambda }$)

A Brownian loop soup of intensity $\lambda >0$ in a planar domain $D$ is a Poisson point process of unrooted Brownian motion loops with intensity measure $\lambda {\mu }_D^{\mathrm{loop}}$, where ${\mu }_D^{\mathrm{loop}}$ is Brownian loop measure restricted to loops contained in $D$. Unrooted means that a cyclic change of starting time does not produce a different loop.

For every measurable family $A$ of loops with ${\mu }_D^{\mathrm{loop}}(A)<\infty$, the number of soup loops in $A$ has Poisson distribution with mean $\lambda {\mu }_D^{\mathrm{loop}}(A)$. Counts in disjoint families are independent. Thus the probability that no loop belongs to $A$ is $\exp (-\lambda {\mu }_D^{\mathrm{loop}}(A))$. Finite restrictions give concrete simulation windows even though the whole intensity is infinite.

The conformal invariance of Brownian loop measure, with the appropriate change of time along each loop, gives conformal invariance of Brownian loop soup. Restricting a soup to loops contained in a smaller domain gives the soup there with the same intensity. This couples random loop geometry to conformal restriction measure on simple loops.^[ref]

Let $f:D\to D^{\prime }$ be a conformal map that is a bijection between planar domains. Mapping every loop of a Brownian loop soup in $D$ by $f$ gives a Brownian loop soup of the same intensity in $D^{\prime }$, after changing the clock along each loop. For a Brownian parametrization $\ell (t)$, the new clock isThe statement concerns unrooted loops: choosing a different starting time on the same loop does not create another object. It follows from conformal invariance of Brownian loop measure and the image property of a Poisson point process.^[ref]

For the dilation $f(z)=az$, spatial lengths multiply by $|a|$ and loop durations by $|a{|}^2$, while the soup intensity parameter stays fixed. Thus the random geometry has no preferred length scale. Keeping the clock change explicit prevents confusing conformal invariance of the loop ensemble with preservation of each loop's original duration.