First Passage Sets in Gaussian Free Fields

• First Passage Sets (FPS) are random closed sets in the two-dimensional Gaussian free field that mirror one-dimensional first passage times by tracking barrier-crossing paths.
• FPS are rigorously defined using local sets, two-valued local sets, and generalized level lines, ensuring key properties like uniqueness and monotonicity.
• They exhibit zero Lebesgue measure, a fractal dimension of 2, and support a nontrivial field measure identified through a Minkowski content gauge with logarithmic correction.

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 (Aru et al., 2017). 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 $r\mapsto |\log r|^{1/2}r^2$ (Aru et al., 2017). 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 (Aru et al., 2018).

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$0 (Aru et al., 2017).

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 $-a$1, the FPS of level $-a$2 is defined as

$-a$3

which directly implements the “path staying above $-a$4” principle (Aru et al., 2018).

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$5 is local for the GFF $-a$6 if, conditionally on $-a$7, the remainder

$-a$8

is a GFF in $-a$9. Writing the harmonic part as $-a$0, one has outside $-a$1

$-a$2

For FPS, the crucial structure is that the field outside the set has boundary condition $-a$3, while the field supported on the set becomes nonnegative after shifting by $-a$4 (Aru et al., 2017).

In the zero-boundary case, the defining properties are that $-a$5 is a local set such that, conditionally on $-a$6, the law of $-a$7 on $-a$8 is that of a GFF with boundary condition $-a$9, and the distribution $r\mapsto |\log r|^{1/2}r^2$0 is a positive measure (Aru et al., 2017). For general bounded harmonic boundary condition $r\mapsto |\log r|^{1/2}r^2$1, the paper formulates the FPS $r\mapsto |\log r|^{1/2}r^2$2 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 (Aru et al., 2017).

In the formulation recalled later, if $r\mapsto |\log r|^{1/2}r^2$3, the conditions simplify to

$r\mapsto |\log r|^{1/2}r^2$4

(Aru et al., 2018). The positivity condition can also be stated distributionally: $r\mapsto |\log r|^{1/2}r^2$5 equivalently $r\mapsto |\log r|^{1/2}r^2$6 for positive test functions $r\mapsto |\log r|^{1/2}r^2$7 (Aru et al., 2018).

The central structural theorem is existence and uniqueness: the FPS $r\mapsto |\log r|^{1/2}r^2$8 of level $r\mapsto |\log r|^{1/2}r^2$9 exists and is unique, and any local set satisfying the defining FPS properties agrees with it almost surely (Aru et al., 2017). The theory also proves monotonicity: $_4$0 (Aru et al., 2017).

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 $_4$1 are bounded-type thin local sets for which the harmonic part only takes the two values $_4$2 and $_4$3 (Aru et al., 2017). The FPS is then obtained by sending the upper level to $_4$4: $_4$5 This limiting procedure expresses the intuition that only the lower barrier $_4$6 is retained in the first-passage construction (Aru et al., 2017).

The implementation relies on generalized level lines, described as SLE$_4$7-type curves coupled with the GFF, and the analysis proves that they remain continuous up to their terminal time even in finitely connected domains (Aru et al., 2017). 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: $_4$8 (Aru et al., 2017). 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 $_4$9 has zero Lebesgue measure, yet its Minkowski dimension is $B_t$0 (Aru et al., 2017). Later work also states the Hausdorff-dimension consequence

$B_t$1

provided the boundary condition is not everywhere $B_t$2 (Aru et al., 2018). The combination of zero area and dimension $B_t$3 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

$B_t$4

is almost surely a nontrivial positive measure, unless the boundary condition is already $B_t$5 everywhere (Aru et al., 2017). Moreover, $B_t$6 is a measurable function of the set $B_t$7 itself (Aru et al., 2017). 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

$B_t$8

More precisely, for every continuous compactly supported $B_t$9,

$-a$0

(Aru et al., 2017). The appearance of the constant $-a$1 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 $-a$2: the set supports a nontrivial measure at exactly the logarithmically corrected $-a$3 scale (Aru et al., 2017).

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 $-a$4, the FPS $-a$5 is a compact optional set, $-a$6 equals $-a$7 on $-a$8, and every connected component of $-a$9 intersects $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$0 (Aru et al., 2018). 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 $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$1, the associated metric graph domains $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$2, and boundary conditions $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$3, 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 $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$4,

$T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$5

in law as $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$6, with convergence of the set component in the Hausdorff topology (Aru et al., 2018). If the fields themselves converge in probability, then the joint convergence also holds in probability (Aru et al., 2018).

The topologies are explicit: closed sets are endowed with the Hausdorff metric, fields converge in $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$7, and domains converge via Hausdorff convergence of complements, implying Carathéodory convergence on connected components (Aru et al., 2018). 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 (Aru et al., 2018).

An additional structural property derived in this framework is local finiteness: for any $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$8, only finitely many connected components of $T_{-a}=\inf\{t\ge 0:\, B_t=-a\}.$9 have diameter larger than $T_{-a}$0 (Aru et al., 2018).

6. Loop soups, excursions, Wick square, and SLE$T_{-a}$1

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 $T_{-a}$2 of intensity $T_{-a}$3 and an independent Poisson point process of boundary-to-boundary excursions $T_{-a}$4. The paper states the polarized isomorphism

$T_{-a}$5

with the sign field $T_{-a}$6 constant on each connected component of the positivity set (Aru et al., 2018). The occupation field identity is

$T_{-a}$7

(Aru et al., 2018).

For $T_{-a}$8 and $T_{-a}$9, 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 (Aru et al., 2018). Passing to the continuum yields the representation

$-a$0

where $-a$1 is a Brownian loop-soup of intensity $-a$2, $-a$3 is a PPP of boundary-to-boundary Brownian excursions, and $-a$4 denotes the closed union of clusters that contain at least one excursion (Aru et al., 2018).

The same paper extends Le Jan’s isomorphism to non-zero boundary conditions: $-a$5 equivalently

$-a$6

(Aru et al., 2018). This couples the FPS representation to the Wick square of the GFF on a common probability space.

The Minkowski-content gauge $-a$7 then acquires an additional interpretation as the natural gauge for the size of critical Brownian loop-soup clusters at $-a$8, equivalently $-a$9 (Aru et al., 2018). The paper notes that for subcritical loop-soups $-a$00, the correct gauge is still unknown (Aru et al., 2018).

FPS are also tied to interfaces converging to SLE$-a$01. For suitable piecewise constant boundary data, the boundary component of $-a$02 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 (Aru et al., 2018). In the simply connected case with constant $-a$03 on one boundary arc, the limit is the trace of an SLE$-a$04 with

$-a$05

(Aru et al., 2018). A particularly simple corollary gives convergence to the Schramm–Sheffield level line, that is, to SLE$-a$06 in $-a$07, in a half-plane-type geometry with boundary values $-a$08 and $-a$09 on the two sides (Aru et al., 2018).

The same framework also yields an FKG property: for non-negative $-a$10, increasing functionals $-a$11 of compact sets satisfy

$-a$12

(Aru et al., 2018).

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 $-a$13: 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 (Sekimoto, 2021). 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 (Godec et al., 2015). In Ornstein–Uhlenbeck dynamics with resetting, first-passage Brownian functionals are path integrals accumulated up to the absorption time $-a$14, and their statistics are reshaped by the competition between confinement and resetting (Dubey et al., 2023).

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 (Aru et al., 2017). 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 (Aru et al., 2017).

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 $-a$15 (Aru et al., 2017, Aru et al., 2018).