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We construct a measure on the thick points of a Brownian loop soup in a
bounded domain D of the plane with given intensity $\theta>0$, which is
formally obtained by exponentiating the square root of its occupation field.
The measure is constructed via a regularisation procedure, in which loops are
killed at a fix rate, allowing us to make use of the Brownian multiplicative
chaos measures previously considered in [BBK94, AHS20, Jeg20a], or via a
discrete loop soup approximation. At the critical intensity $\theta = 1/2$, it
is shown that this measure coincides with the hyperbolic cosine of the Gaussian
free field, which is closely related to Liouville measure. This allows us to
draw several conclusions which elucidate connections between Brownian
multiplicative chaos, Gaussian free field and Liouville measure. For instance,
it is shown that Liouville-typical points are of infinite loop multiplicity,
with the relative contribution of each loop to the overall thickness of the
point being described by the Poisson–Dirichlet distribution with parameter
$\theta = 1/2$. Conversely, the Brownian chaos associated to each loop
describes its microscopic contribution to Liouville measure. Along the way, our
proof reveals a surprising exact integrability of the multiplicative chaos
associated to a killed Brownian loop soup. We also obtain some estimates on the
discrete and continuous loop soups which may be of independent interest.

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