Probability Seminar

For each central charge $c\in (0,1]$, we construct a conformally invariant field which is a measurable function of the local time field $\mathcal{L}$ of the Brownian loop soup with intensity $c$ and i.i.d. signs given to each cluster. This field is canonically associated to $\mathcal{L}$, in a sense which is similar to the isomorphism theory that associates the Gaussian free field to the loop soup with critical intensity. Isomorphisms between Brownian motions and random fields were previously developed by Symanzik, Brydges-Fröhlich-Spencer, Dynkin and Le Jan in several different settings.

In a key intermediate step, we obtain the crossing exponent for the event that a cluster in the subcritical loop soup passes near a given point. Among other things, it allows us to deduce that the Minkowski gauge function of a cluster in a loop soup with intensity $c\in (0,1)$, which is $t^2 f(t)$ for some $f(t)=|\log t|^{1-c/2+o(1)}$.

In the end, I will present several conjectures about this family of fields. This talk is based on joint works in progress with Antoine Jego (EPFL) and Titus Lupu (CNRS).