  This article deals with limit theorems for certain loop variables for loop
soups whose intensity approaches infinity. We first consider random walk loop
soups on finite graphs and obtain a central limit theorem when the loop
variable is the sum over all loops of the integral of each loop against a given
one-form on the graph. An extension of this result to the noncommutative case
of loop holonomies is also discussed. As an application of the first result, we
derive a central limit theorem for windings of loops around the faces of a
planar graphs. More precisely, we show that the winding field generated by a
random walk loop soup, when appropriately normalized, has a Gaussian limit as
the loop soup intensity tends to $\infty$, and we give an explicit formula for
the covariance kernel of the limiting field. We also derive a Spitzer-type law
for windings of the Brownian loop soup, i.e., we show that the total winding
around a point of all loops of diameter larger than $\delta$, when multiplied
by $1/\log\delta$, converges in distribution to a Cauchy random variable as
$\delta \to 0$.
