The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

Abstract

In a previous article, we introduced the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to $-a$. This description can be taken as a definition of the FPS for the metric graph GFF, and in the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff distance. We also draw numerous consequences; in particular, we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph GFF converge to $\hbox {SLE}_4$ level lines. These results improve our understanding of the continuum GFF, by strengthening its relationship with the critical Brownian loop-soup. Indeed, a new construction of the FPS using clusters of Brownian loops and excursions helps to strengthen the known GFF isomorphism theorems, and allows us to use Brownian loop-soup techniques to prove technical results on the geometry of the GFF. We also obtain a new representation of Brownian loop-soup clusters, and as a consequence, we prove that the clusters of a critical Brownian loop-soup admit a non-trivial Minkowski content in the gauge $r\mapsto |\log r|^{1/2}r^2$.

Appendix

1.1 Proof of Proposition 2.3

If $u\equiv 0$, there are no excursions we are in the setting of Le Jan’s isomorphism for loop-soups [38, 40]. If u is constant and strictly positive, then the proposition follows by combining Le Jan’s isomorphism and the generalized second Ray-Knight theorem [45, 68]. Indeed, then one can consider the whole boundary $\partial \mathcal {G}$ as a single vertex, and the boundary to boundary excursions as excursions outside this vertex.

The case of u non-constant can be reduced to the previous one. We first assume that u is strictly positive on $\partial \mathcal {G}$. The general case can be obtained by taking the limit. We define new conductances on the edges:

$$\begin{aligned} \widehat{C}(x,y):=C(x,y)u(x)u(y), \end{aligned}$$

where x and y are neighbours in $\mathcal {G}$. Let $\hat{\phi }$ be the 0 boundary GFF associated to the new conductances $\widehat{C}$. We claim that

$$\begin{aligned} (\hat{\phi }(x))_{x\in V} {\mathop {=}\limits ^{(d)}} (u(x)^{-1}\phi (x))_{x\in V}. \end{aligned}$$

To check the identity in law it is sufficient to show the equality of energy functions, since the latter give the densities (2.1). Let $(f(x))_{x\in \mathcal G}$ be such that for all $x\in \partial \mathcal G$, $f(x)=0$ and compute

$$\begin{aligned} \mathcal {E}_{\mathcal {G}}(uf,uf)= & {} -\sum _{x\in V}\sum _{y\sim x} u(x)f(x)C(x,y)(u(y)f(y)-u(x)f(x))\\= & {} -\sum _{x\in V}\sum _{y\sim x} u(x)f(x)C(x,y)u(y)(f(y)-f(x))\\&+\sum _{x\in V}\sum _{y\sim x} C(x,y)(u(y)-u(x))u(x)f(x)^{2}\\= & {} -\sum _{x\in V}\sum _{y\sim x} f(x)\widehat{C}(x,y)(f(y)-f(x))\\&+ \sum _{x\in \partial \mathcal {G}}\sum _{y\sim x} C(x,y)(u(y)-u(x))u(x)f(x)^{2}\\= & {} \widehat{\mathcal {E}}(f,f)+0. \end{aligned}$$

From the second to the third line we used that u is harmonic.

Now, we can apply the case of constant boundary conditions to ${\frac{1}{2}(\hat{\phi }+1)^{2}}$. We get that it is distributed like the occupation field of a loop-soup of parameter $\alpha =1/2$ and an independent Poissonian family of excursions from $\hat{x}$ to $\hat{x}$, both associated to the jump rates $\widehat{C}(x,y)$. If on these paths we perform the time change

$$\begin{aligned} dt=u(x)^{-2} ds, \end{aligned}$$

(5.2)

we get $\mathcal {L}^\mathcal {G}_{1/2}$ and $\varXi _{u}^{\mathcal {G}}$. The time change (5.2) multiplies the occupation field by $u^{2}$, which exactly transforms $(\hat{\phi }+1)^{2}$ into $(\phi +u)^{2}$.