Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time

Abstract

In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field $\Lambda$: When conditioned on $\Lambda$, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve $\Lambda$ impose an exceptional behaviour of all the loops when they go through these points.