Exact probabilities for some topological events for metric graph GFF

Date: 2022-08-08

Time: 14:00 - 15:00


Titus Lupu


The metric graph GFF is obtained by interpolating the discrete GFF by Brownian bridges inside the edges. The metric graph GFF satisfies more exact solvability properties than the discrete GFF. In particular, I will present formulas for probabilities for some topological events on the sign clusters of the GFF on metric graphs. More precisely this is related to the Z/2Z-homology of these sign clusters. These probabilities are square roots of ratios of two determinants of Laplacians, one being the usual graph Laplacian, and the other one being a twisted Laplacian. As an example, one gets on planar annular-shapped (i.e. two-connected) graphs the probability of existence of a non-contractible sign cluster that separates the inner and the outer boundary. In the scaling limit, this particular probability agrees with the one that can be computed through the level lines of the 2D continuum GFF.