= Brownian loop soup
{c}
{title2=$\mathcal L_\lambda$}
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 $\lambda\mu_D^{\rm loop}$, where $\mu_D^{\rm loop}$ 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 $\mu_D^{\rm loop}(A)<\infty$, the number of soup loops in $A$ has <Poisson distribution> with mean $\lambda\mu_D^{\rm loop}(A)$. Counts in disjoint families are independent. Thus the probability that no loop belongs to $A$ is $\exp(-\lambda\mu_D^{\rm loop}(A))$. 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>.<The Brownian loop soup by Gregory F. Lawler and Wendelin Werner>[Sections 4.1 and 5.3, loop measure and the Poissonian soup construction]{ref}
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