---
title: First Passage Sets in Gaussian Free Fields
url: https://www.emergentmind.com/topics/first-passage-sets-fps
type: topic
---

# First Passage Sets in Gaussian Free Fields

Searching arXiv for the specified FPS literature and closely related first-passage works.
First Passage Sets (FPS) are random closed sets associated with the two-dimensional continuum Gaussian free field (GFF) that formalize a geometric analogue of one-dimensional first hitting times. Informally, the FPS of level \(-a\) is the set of points in a domain that can be connected to the boundary by a path along which the field does not go below \(-a\), making it the two-dimensional analogue of the first hitting time of \(-a\) by Brownian motion [1706.07737]. In the continuum theory, this heuristic is made precise through the framework of local sets: an FPS is characterized by a boundary-value condition on the complement together with a positivity property of the field restricted to the set. The foundational results establish existence, uniqueness, monotonicity, fractal geometry, and an identification of the field carried by the FPS with a Minkowski content measure in the gauge \(r\mapsto |\log r|^{1/2}r^2\) [1706.07737]. Subsequent work shows that metric-graph FPS converge to continuum FPS in the Hausdorff metric and connects FPS to Brownian loop soups, Brownian excursions, Wick square isomorphisms, and SLE\(_4\)-type interfaces [1805.09204].

## 1. Brownian first passage and the GFF analogue

The motivating analogy begins with one-dimensional Brownian motion. For a Brownian motion \(B_t\), the first passage time of level \(-a\) is
\[
T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.
\]
Before time \(T_{-a}\), the Brownian path remains above \(-a\). The FPS transfers this idea from temporal first hitting to a spatial-geometric object in a random field: points belong to the FPS if they remain connected to the boundary through locations where the GFF stays above the barrier \(-a\) [1706.07737].

This analogy is only heuristic in the continuum because the GFF is not pointwise defined. On metric graphs, however, the pathwise description is literal. For the metric graph GFF \(\tilde\phi+u\), the FPS of level \(-a\) is defined as
\[
\widetilde{}_{-a}^{u}(\tilde{\phi}):= \left\{x\in \widetilde{\mathcal G}\ \middle|\ \exists \gamma \text{ continuous path from }x\text{ to }\partial\mathcal G\text{ such that }\tilde\phi\ge -a\text{ on }\gamma\right\},
\]
which directly implements the “path staying above \(-a\)” principle [1805.09204].

A plausible implication is that the term “first passage” in this context refers not to a stopping time but to a maximal connected region preserving accessibility to the boundary above a prescribed level. The continuum theory reconstructs this intuition through conditional independence and harmonic decomposition rather than through pointwise inequalities.

## 2. Local-set formulation and axiomatic characterization

The rigorous continuum definition is expressed in terms of local sets of the GFF. A random closed set \(A\subset \overline D\) is local for the GFF \(\Phi\) if, conditionally on \((A,\Phi_A)\), the remainder
\[
\Phi^A := \Phi-\Phi_A
\]
is a GFF in \(D\setminus A\). Writing the harmonic part as \(h_A\), one has outside \(A\)
\[
\Phi = h_A + \Phi^A.
\]
For FPS, the crucial structure is that the field outside the set has boundary condition \(-a\), while the field supported on the set becomes nonnegative after shifting by \(a\) [1706.07737].

In the zero-boundary case, the defining properties are that \({}_{-a}\) is a local set such that, conditionally on \({}_{-a}\), the law of \(\Phi\) on \(D\setminus {}_{-a}\) is that of a GFF with boundary condition \(-a\), and the distribution \(\Phi_{{}_{-a}}+a\) is a positive measure [1706.07737]. For general bounded harmonic boundary condition \(u\), the paper formulates the FPS \({}_{-a}^u\) by four requirements: a harmonic boundary-value condition on each connected component of the complement, positivity of the residual field, a compatibility condition near boundary points, and exclusion of isolated points together with a mild separation condition for connected components [1706.07737].

In the formulation recalled later, if \(u\ge -a\), the conditions simplify to
\[
h_{{}_{-a}^{u}}+u=-a \quad \text{in } D\setminus {}_{-a}^{u},\qquad \Phi_{{}_{-a}^{u}}+u+a\ge 0
\]
[1805.09204]. The positivity condition can also be stated distributionally:
\[
\Phi_{{}_{-a}^{u}} - h_{{}_{-a}^{u}} \ge 0,
\]
equivalently \((\Phi_{{}_{-a}^{u}}-h_{{}_{-a}^{u}},f)\ge 0\) for positive test functions \(f\) [1805.09204].

The central structural theorem is existence and uniqueness: the FPS \({}_{-a}^u\) of level \(-a\) exists and is unique, and any local set satisfying the defining FPS properties agrees with it almost surely [1706.07737]. The theory also proves monotonicity:
\[
a\le a',\ u\le u' \quad\Longrightarrow\quad {}_{-a}^u\subseteq {}_{-a'}^{u'} \quad \text{a.s.}
\]
[1706.07737].

## 3. Construction from two-valued local sets and level lines

The continuum construction proceeds through two-valued local sets and generalized level lines. The two-valued local sets \({}_{-a,b}^u\) are bounded-type thin local sets for which the harmonic part only takes the two values \(-a\) and \(b\) [1706.07737]. The FPS is then obtained by sending the upper level to \(+\infty\):
\[
{}_{-a}^u = \overline{\bigcup_{n\in \mathbb N} {}_{-a,n}^u}.
\]
This limiting procedure expresses the intuition that only the lower barrier \(-a\) is retained in the first-passage construction [1706.07737].

The implementation relies on generalized level lines, described as SLE\(_4(\rho)\)-type curves coupled with the GFF, and the analysis proves that they remain continuous up to their terminal time even in finitely connected domains [1706.07737]. These level lines are used to build two-valued local sets and therefore the FPS component by component.

The relation between FPS and two-valued sets is especially transparent in simply connected domains:
\[
{}_{-a,b} = {}_{-a}\cap {}_{b}
\]
[1706.07737]. This identity places FPS within the broader local-set hierarchy generated by level-line constructions.

A plausible implication is that FPS should be viewed not as an isolated object but as the one-sided limit of a symmetric two-threshold theory. In that sense, the FPS interpolates between level-line geometry and positivity properties of the field mass carried by local sets.

## 4. Geometric structure and field carried by the set

The geometric theory shows that FPS are fractal and large in a metric sense, while remaining negligible in area. The set \({}_{-a}^u\) has zero Lebesgue measure, yet its Minkowski dimension is \(2\) [1706.07737]. Later work also states the Hausdorff-dimension consequence
\[
\dim_H({}_{-a}^{u})=2 \quad \text{a.s.}
\]
provided the boundary condition is not everywhere \(\le -a\) [1805.09204]. The combination of zero area and dimension \(2\) is a defining feature of the set’s critical geometry.

Unlike many thin local sets, FPS is not thin: the GFF charges it. The measure
\[
\nu_{{}_{-a}^{u}}:=\Phi_{{}_{-a}^{u}}-h_{{}_{-a}^{u}}
\]
is almost surely a nontrivial positive measure, unless the boundary condition is already \(\le -a\) everywhere [1706.07737]. Moreover, \(\Phi_{{}_{-a}^{u}}\) is a measurable function of the set \({}_{-a}^{u}\) itself [1706.07737]. This measurability result means that the geometry of the FPS determines the associated field contribution supported on the set.

One of the main theorems identifies this measure with a Minkowski content measure in the gauge
\[
r\mapsto |\log r|^{1/2}r^2.
\]
More precisely, for every continuous compactly supported \(f\),
\[
\nu_{{}_{-a}^{u}}
=
\lim_{r\to 0}\frac12 |\log r|^{1/2}
\int_D f(z)\,\mathbf 1_{\{d(z,{}_{-a}^{u})\le r\}}\,dz
\]
[1706.07737]. The appearance of the constant \(1/2\) is part of the identification stated in the theorem.

This establishes a direct equivalence between a field-theoretic object and a renormalized geometric neighborhood volume. The result also explains why the Minkowski dimension is \(2\): the set supports a nontrivial measure at exactly the logarithmically corrected \(r^2\) scale [1706.07737].

## 5. Metric-graph approximation and convergence to the continuum

The pathwise definition of FPS on metric graphs provides a discrete-continuum bridge. On the metric graph \(\widetilde{\mathcal G}\), the FPS \(\widetilde{}_{-a}^{u}\) is a compact optional set, \(\tilde\phi+u\) equals \(-a\) on \(\partial \widetilde{}_{-a}^{u}\setminus \partial\mathcal G\), and every connected component of \(\widetilde{}_{-a}^{u}\) intersects \(\partial\mathcal G\) [1805.09204]. These properties make the metric-graph model a literal realization of the heuristic path-based picture.

The main convergence theorem states that, under natural assumptions on the approximating domains \(D_n\), the associated metric graph domains \(\widetilde D_n\), and boundary conditions \(u_n\to u\), the pair consisting of the metric graph GFF and the metric graph FPS converges to the continuum GFF and continuum FPS. For each connected component \(D^z\),
\[
\bigl(\widehat \phi_n^{D^z},\, (\widetilde {}_{-a}^{u_n}\cap D^z)\cup \partial D^z\bigr) \ \Rightarrow\ \bigl(\Phi^{D^z},\, {}_{-a}^{u}\bigr)
\]
in law as \(n\to\infty\), with convergence of the set component in the Hausdorff topology [1805.09204]. If the fields themselves converge in probability, then the joint convergence also holds in probability [1805.09204].

The topologies are explicit: closed sets are endowed with the Hausdorff metric, fields converge in \(H^{-1-}([-C,C]^2)\), and domains converge via Hausdorff convergence of complements, implying Carathéodory convergence on connected components [1805.09204]. The proof uses convergence of the metric graph GFF to the continuum GFF, convergence of metric local sets to continuum local sets, uniqueness of FPS, and a Beurling estimate for the limiting boundary condition [1805.09204].

An additional structural property derived in this framework is local finiteness: for any \(\varepsilon>0\), only finitely many connected components of \(D\setminus {}_{-a}^{u}\) have diameter larger than \(\varepsilon\) [1805.09204].

## 6. Loop soups, excursions, Wick square, and SLE\(_4\)

A major consequence of the metric-graph approximation is a Poissonian representation of FPS in terms of Brownian loops and excursions. On the metric graph, one considers a loop-soup \(\mathcal L^{\widetilde{\mathcal G}}_{1/2}\) of intensity \(1/2\) and an independent Poisson point process of boundary-to-boundary excursions \(\Xi^{\widetilde{\mathcal G}}_u\). The paper states the polarized isomorphism
\[
\left(\sigma(x)\sqrt{2}\left(L^{x}(\mathcal{L}^{\widetilde{\mathcal G}}_{1/2}) +L^{x}(\Xi_{u}^{\widetilde{\mathcal G}})\right)^{1/2}\right)_{x\in \widetilde{\mathcal G}}
\stackrel{d}{=} \tilde\phi+u,
\]
with the sign field \(\sigma(x)\in\{-1,1\}\) constant on each connected component of the positivity set [1805.09204]. The occupation field identity is
\[
\left(L^{x}(\mathcal{L}^{\mathcal G}_{1/2}) +L^{x}(\Xi_{u}^{\mathcal G})}\right)_{x\in V\setminus\partial\mathcal G}
\stackrel{d}{=} \left(\frac12(\phi+u)^2(x)\right)_{x\in V\setminus\partial\mathcal G}
\]
[1805.09204].

For \(a=0\) and \(u\ge 0\), the metric-graph FPS is exactly the union of the topological closures of all clusters of loops and excursions that contain at least one excursion, together with the boundary [1805.09204]. Passing to the continuum yields the representation
\[
\mathcal{A}(\mathcal{L}^{D}_{1/2},\Xi^{D}_{u})\cup \partial D \stackrel{d}{=} {}_{0}^{u},
\]
where \(\mathcal{L}^{D}_{1/2}\) is a Brownian loop-soup of intensity \(1/2\), \(\Xi^{D}_{u}\) is a PPP of boundary-to-boundary Brownian excursions, and \(\mathcal A(\cdot)\) denotes the closed union of clusters that contain at least one excursion [1805.09204].

The same paper extends Le Jan’s isomorphism to non-zero boundary conditions:
\[
L_{\rm ctr}(\mathcal{L}^{D}_{1/2})+L(\Xi^{D}_{u}) \stackrel{d}{=} \frac12 :\Phi^{2}: + u\Phi + \frac12 u^{2},
\]
equivalently
\[
L_{\rm ctr}(\mathcal{L}^{D}_{1/2})+L_{\rm ctr}(\Xi^{D}_{u}) \stackrel{d}{=} \frac12 :(\Phi+u)^{2}:
\]
[1805.09204]. This couples the FPS representation to the Wick square of the GFF on a common probability space.

The Minkowski-content gauge \(r\mapsto |\log r|^{1/2}r^2\) then acquires an additional interpretation as the natural gauge for the size of critical Brownian loop-soup clusters at \(c=1\), equivalently \(\alpha=\tfrac12\) [1805.09204]. The paper notes that for subcritical loop-soups \((c<1)\), the correct gauge is still unknown [1805.09204].

FPS are also tied to interfaces converging to SLE\(_4\). For suitable piecewise constant boundary data, the boundary component of \({}_{-\lambda}^{u}\) separating prescribed boundary arcs is the generalized level line, and corresponding metric graph interfaces converge in law in the Hausdorff topology to this continuum curve [1805.09204]. In the simply connected case with constant \(u=b>-\lambda\) on one boundary arc, the limit is the trace of an SLE\(_4(\rho)\) with
\[
\rho = b/\lambda - 1
\]
[1805.09204]. A particularly simple corollary gives convergence to the Schramm–Sheffield level line, that is, to SLE\(_4\) in \(\mathbb H\), in a half-plane-type geometry with boundary values \(-\lambda\) and \(+\lambda\) on the two sides [1805.09204].

The same framework also yields an FKG property: for non-negative \(u\), increasing functionals \(F_1,F_2\) of compact sets satisfy
\[
\mathbb E[F_1({}_{0}^{u})F_2({}_{0}^{u})] \ge \mathbb E[F_1({}_{0}^{u})]\mathbb E[F_2({}_{0}^{u})]
\]
[1805.09204].

## 7. Conceptual scope and relation to first-passage theory

FPS belong to a broader family of first-passage concepts, but their role is distinct. In finite Markov networks, first passage is formulated as a probability-flux problem into an absorbing target set \(B\): the first passage density is the loss rate of the survival probability on the reduced non-absorbing network, and the target is realized as a sink of probability [2110.02216]. In heterogeneous diffusion, the full first-passage-time distribution reveals direct, intermediate, and reflected trajectory regimes that are not captured by the mean first-passage time alone [1510.00932]. In Ornstein–Uhlenbeck dynamics with resetting, first-passage Brownian functionals are path integrals accumulated up to the absorption time \(t_f\), and their statistics are reshaped by the competition between confinement and resetting [2304.05226].

These works illuminate the terminology shared by FPS, but the GFF construction is geometric rather than temporal. The Brownian analogy is specifically to first hitting of a barrier, while the two-dimensional object is a random set extracted from the field by a local-set decomposition [1706.07737]. A plausible implication is that FPS should not be conflated with first-passage times of a Markov process: the common theme is the barrier-crossing structure, but the mathematical realization differs fundamentally. In the GFF setting, the outcome is a fractal set carrying a positive measure and encoding both connectivity and field mass [1706.07737].

The current theory therefore situates FPS at the intersection of local-set theory, critical planar probability, Gaussian multiplicative chaos, and loop-soup isomorphisms. Its principal achievements are the unique characterization of FPS as local sets, their construction from two-valued local sets and level lines, their convergence from metric graphs to the continuum, and the identification of their field content with Minkowski content in the gauge \(r\mapsto |\log r|^{1/2}r^2\) [1706.07737; 1805.09204].

Source: https://www.emergentmind.com/topics/first-passage-sets-fps