Excursion Clusters in Random Field Theory

• Excursion clusters are connected components derived from threshold exceedance sets in random fields and tree-indexed processes, encompassing discrete and continuum frameworks.
• They form the basis for probabilistic decompositions using tools like local sets, Minkowski content measures, and restricted Wick powers in 2D Gaussian free fields.
• Their study enables the derivation of exact size laws, boundary genealogies, and neighborhood asymptotic behaviors in both lattice GFF and tree-indexed models.

Excursion clusters are connected components associated with excursion events of random fields and tree-indexed processes. In the literature considered here, the term covers several closely related objects: connected components of a superlevel set $\{X_t>u\}$ for a discrete random field, connected components of the complement of a zero set for tree-indexed Brownian or Markov processes, and, in the two-dimensional continuum Gaussian free field (GFF), sign components constructed through local-set, $\mathrm{CLE}_4$, and Brownian loop-soup methods. In the 2D continuum GFF, recent work makes the notion especially concrete: the field admits an excursion decomposition into countably many signed Minkowski content measures supported on random compact connected sets, and these clusters support restricted Wick powers and control small-neighborhood asymptotics (Aru et al., 2023, Lupu, 1 Sep 2025).

1. Terminology and model-dependent meanings

In lattice random-field theory, an excursion cluster is typically a connected component of an excursion set. For a discrete field $\{X_t:t\in\mathbb Z^d\}$ and threshold $u\in\mathbb R$, the excursion set is

$E_u=\{t\in\mathbb Z^d:X_t>u\},$

and clusters are its connected components under a chosen neighborhood system, such as nearest-neighbor or Moore connectivity. Their size is the number of lattice points they contain (Cheng et al., 21 Jan 2026).

In tree-indexed settings, the same term refers to connected components of the complement of a distinguished state. For Brownian motion indexed by the Brownian tree, excursion clusters are the connected components of

$\{u\in\mathcal T_\zeta:V_u\neq 0\},$

and each such component is itself a continuous tree equipped with labels (Abraham et al., 2015). For Markov processes indexed by Lévy trees, excursions away from a regular and instantaneous point $x$ are the connected subtree-components of

$\{a\in\mathcal T_H:\xi_a\neq x\},$

and these are indexed by a tree-local time at $x$ (Riera et al., 2024).

In the 2D continuum GFF, excursion clusters are the sign components of the field. Because the field is a distribution rather than a pointwise-defined function, these clusters are not defined as literal connected components of $\{\Phi>0\}$ or $\mathrm{CLE}_4$0. Instead, they are constructed through local sets, $\mathrm{CLE}_4$1, first passage sets, and Brownian loop-soup geometry (Aru et al., 2023).

2. Excursion clusters of the two-dimensional continuum GFF

The standard planar setup assumes an open, bounded, connected, and simply connected domain $\mathrm{CLE}_4$2, together with a continuum GFF $\mathrm{CLE}_4$3 whose covariance is the Dirichlet Green function

$\mathrm{CLE}_4$4

where $\mathrm{CLE}_4$5 is the conformal radius (Lupu, 1 Sep 2025).

Aru–Lupu–Sepúlveda constructed a countable family of random compact connected sets

$\mathrm{CLE}_4$6

in decreasing order of diameter, each carrying a sign

$\mathrm{CLE}_4$7

These are the excursion clusters, also called excursion sets or sign clusters. Conditionally on the clusters $\mathrm{CLE}_4$8, the signs $\mathrm{CLE}_4$9 are i.i.d. uniform $\{X_t:t\in\mathbb Z^d\}$0. Each cluster carries a positive finite measure $\{X_t:t\in\mathbb Z^d\}$1, its Minkowski content measure, obtained as the almost sure weak limit of

$\{X_t:t\in\mathbb Z^d\}$2

The field decomposes as

$\{X_t:t\in\mathbb Z^d\}$3

with convergence in $\{X_t:t\in\mathbb Z^d\}$4 for every $\{X_t:t\in\mathbb Z^d\}$5 (Lupu, 1 Sep 2025).

This decomposition is one of the clearest manifestations that excursion clusters are non-thin local sets. They have zero Lebesgue measure, but the field restricts nontrivially to them. For an excursion cluster, the restriction is $\{X_t:t\in\mathbb Z^d\}$6, positive or negative according to the sign. In the broader excursion decomposition of the zero-boundary 2D GFF, the law of $\{X_t:t\in\mathbb Z^d\}$7 coincides with the law of the topological closures of clusters of the critical Brownian loop soup at intensity $\{X_t:t\in\mathbb Z^d\}$8, and the outer boundaries of outermost clusters form $\{X_t:t\in\mathbb Z^d\}$9 loops (Aru et al., 2023).

A further structural fact is that each excursion cluster is conditionally a first passage set. If $u\in\mathbb R$0 is the outer boundary of $u\in\mathbb R$1, then, conditionally on the exterior information consisting of earlier clusters, $u\in\mathbb R$2, and $u\in\mathbb R$3, the field

$u\in\mathbb R$4

is distributed as a GFF in $u\in\mathbb R$5 with boundary value $u\in\mathbb R$6 on $u\in\mathbb R$7, where $u\in\mathbb R$8 is the Schramm–Sheffield height gap, and $u\in\mathbb R$9 is the first passage set of this conditional field from level $E_u=\{t\in\mathbb Z^d:X_t>u\},$0 down to level $E_u=\{t\in\mathbb Z^d:X_t>u\},$1 (Lupu, 1 Sep 2025).

3. Wick powers and restriction theory on excursion clusters

The paper on Wick powers and excursion clusters studies how renormalized powers of the continuum GFF interact with these fractal supports. For an individual cluster $E_u=\{t\in\mathbb Z^d:X_t>u\},$2, it defines

$E_u=\{t\in\mathbb Z^d:X_t>u\},$3

so that, conditionally on $E_u=\{t\in\mathbb Z^d:X_t>u\},$4, $E_u=\{t\in\mathbb Z^d:X_t>u\},$5 is a GFF on $E_u=\{t\in\mathbb Z^d:X_t>u\},$6 with boundary condition $E_u=\{t\in\mathbb Z^d:X_t>u\},$7. The cluster-restricted Wick fields are

$E_u=\{t\in\mathbb Z^d:X_t>u\},$8

where the Wick powers are renormalized using the Green function of $E_u=\{t\in\mathbb Z^d:X_t>u\},$9, not that of the ambient domain (Lupu, 1 Sep 2025).

A direct restriction formula uses the variance defect

$\{u\in\mathcal T_\zeta:V_u\neq 0\},$0

together with its regularized version $\{u\in\mathcal T_\zeta:V_u\neq 0\},$1. The theorem states that for every $\{u\in\mathcal T_\zeta:V_u\neq 0\},$2,

$\{u\in\mathcal T_\zeta:V_u\neq 0\},$3

with convergence in $\{u\in\mathcal T_\zeta:V_u\neq 0\},$4 for every $\{u\in\mathcal T_\zeta:V_u\neq 0\},$5. A second representation writes $\{u\in\mathcal T_\zeta:V_u\neq 0\},$6 as a limit of Gaussian multiplicative chaos germs living outside the cluster (Lupu, 1 Sep 2025).

The central structural fact is an odd/even dichotomy. For odd $\{u\in\mathcal T_\zeta:V_u\neq 0\},$7, $\{u\in\mathcal T_\zeta:V_u\neq 0\},$8 is a generalized function supported on $\{u\in\mathcal T_\zeta:V_u\neq 0\},$9. Thus the odd Wick powers admit bona fide restrictions to excursion clusters. For even $x$0, the situation is different: $x$1 extends to $x$2, where it coincides with the explicit smooth function

$x$3

Because $x$4 as $x$5, this blow-up is non-integrable near the cluster. Even Wick powers therefore do not restrict cleanly to the cluster as distributions supported on $x$6; they require a singular compensating term living off the cluster (Lupu, 1 Sep 2025).

This distinction has a global counterpart. Odd Wick powers decompose naturally into signed contributions attached cluster-by-cluster, whereas even Wick powers carry additional off-cluster terms supported in the interiors bounded by cluster outer boundaries. The decomposition therefore mirrors the same restriction/non-restriction dichotomy already visible for a single cluster (Lupu, 1 Sep 2025).

4. Neighborhood asymptotics and geometric reconstruction

For an excursion cluster $x$7, the $x$8-neighborhoods are defined in conformal-radius geometry by

$x$9

and

$\{a\in\mathcal T_H:\xi_a\neq x\},$0

These neighborhoods admit $\{a\in\mathcal T_H:\xi_a\neq x\},$1-asymptotic expansions into half-integer powers

$\{a\in\mathcal T_H:\xi_a\neq x\},$2

and the coefficients are exactly the restricted odd Wick powers $\{a\in\mathcal T_H:\xi_a\neq x\},$3. The even Wick powers do not appear in these expansions (Lupu, 1 Sep 2025).

The leading coefficient recovers the cluster Minkowski content measure. Since $\{a\in\mathcal T_H:\xi_a\neq x\},$4, the first term is

$\{a\in\mathcal T_H:\xi_a\neq x\},$5

Thus the neighborhood geometry “sees” the measure carried by the cluster itself before it sees higher odd Wick restrictions (Lupu, 1 Sep 2025).

The same work gives a multiscale inversion formula: by combining indicator functions of neighborhoods at several scales $\{a\in\mathcal T_H:\xi_a\neq x\},$6, one can recover $\{a\in\mathcal T_H:\xi_a\neq x\},$7 as a limit in $\{a\in\mathcal T_H:\xi_a\neq x\},$8. This makes the odd restricted fields geometrically reconstructible from the cluster’s small-scale conformal-radius neighborhoods, rather than merely abstract conditional expectations (Lupu, 1 Sep 2025).

The paper compares these expansions with Le Gall’s expansion for the planar Wiener sausage. The analogy is structural, but the exponents differ. For excursion clusters and first passage sets, the powers are half-integer, while for the Wiener sausage they are integer,

$\{a\in\mathcal T_H:\xi_a\neq x\},$9

The paper suggests that this discrepancy reflects the accumulation of arbitrarily small Brownian loops in the loop-soup cluster representation (Lupu, 1 Sep 2025).

5. Discrete random fields and exact size laws

In discrete random fields, excursion clusters are connected components of threshold exceedance sets. For

$x$0

clusters are finite connected components under the chosen connectivity rule. In one dimension this reduces to runs of consecutive exceedances; in higher dimensions it produces connected excursion regions. The stationary theory canonically roots each finite cluster at its lexicographically smallest site (Cheng et al., 21 Jan 2026).

The one-dimensional stationary case admits an exact formula. If $x$1 is the cluster size above threshold $x$2, then for $x$3,

$x$4

For i.i.d. noise with $x$5 and $x$6, this becomes the geometric law

$x$7

In higher dimensions, the exact stationary cluster-size distribution is

$x$8

where the sum runs over connected rooted cluster shapes $x$9 of size $\{\Phi>0\}$0 and $\{\Phi>0\}$1 is the exterior neighbor set (Cheng et al., 21 Jan 2026).

The same paper distinguishes several nonequivalent cluster laws: rooted-cluster distributions, origin-containing distributions, and peak-based cluster-size distributions. In nonstationary fields, it proposes the peak-based law

$\{\Phi>0\}$2

which remains tractable because it is local in $\{\Phi>0\}$3. The framework applies to Gaussian and non-Gaussian fields and depends only on finite-dimensional joint probabilities such as

$\{\Phi>0\}$4

This is a different notion from continuum GFF sign clusters, but it preserves the common core idea of connected exceedance components (Cheng et al., 21 Jan 2026).

6. Tree-indexed excursion clusters and boundary genealogies

For Brownian motion indexed by the Brownian tree, excursion clusters are connected components of the complement of the zero set. Each component is itself a compact continuous real tree with a continuous label function. Their boundary lengths are encoded by a random variable $\{\Phi>0\}$5, and the collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism

$\{\Phi>0\}$6

Conditionally on the boundary lengths, the different excursions are independent, and their conditional law is governed by the excursion measure $\{\Phi>0\}$7, the analogue of Itô’s excursion measure (Abraham et al., 2015).

For Markov processes indexed by Lévy trees, the analogue is formulated relative to a regular and instantaneous point $\{\Phi>0\}$8 of the state space. Excursion clusters are the connected subtree-components of

$\{\Phi>0\}$9

The tree-local time $\mathrm{CLE}_4$00 at $\mathrm{CLE}_4$01 indexes the excursions, and the point measure

$\mathrm{CLE}_4$02

is a Poisson point measure with intensity $\mathrm{CLE}_4$03. Collapsing each excursion component to a point yields another Lévy tree, the tree coded by the local time, which records the genealogy of the excursion clusters themselves (Riera et al., 2024).

These tree-indexed theories clarify an important structural point. In one-parameter excursion theory, an excursion is supported on an interval. In the tree-indexed setting, an excursion cluster is already a branching connected component, and the collection of clusters has its own genealogical organization. The 2D continuum GFF decomposition is formally different, but it shares the same broad pattern: excursion clusters carry intrinsic measures, are not mere pointwise sign domains, and admit a probabilistic decomposition theory (Abraham et al., 2015, Riera et al., 2024).

7. Radius tails and off-critical geometry in lattice GFF

For the discrete GFF on $\mathrm{CLE}_4$04, $\mathrm{CLE}_4$05, excursion clusters are pointwise-defined connected components of

$\mathrm{CLE}_4$06

The radius of the finite cluster containing the origin is encoded by the truncated one-arm event

$\mathrm{CLE}_4$07

Recent work proves a sharp dimensional dichotomy for its tail (Goswami et al., 2021).

In dimension $\mathrm{CLE}_4$08,

$\mathrm{CLE}_4$09

for every $\mathrm{CLE}_4$10. Thus the tail is sub-exponential in $\mathrm{CLE}_4$11, of principal exponential order

$\mathrm{CLE}_4$12

In dimensions $\mathrm{CLE}_4$13, for every $\mathrm{CLE}_4$14, there exist $\mathrm{CLE}_4$15 such that

$\mathrm{CLE}_4$16

The same work extends the dichotomy to truncated two-point functions and to the two-arms probability for annuli crossings (Goswami et al., 2021).

The mechanism is capacity-driven. For the line segment

$\mathrm{CLE}_4$17

the capacity satisfies

$\mathrm{CLE}_4$18

This explains why large finite excursion clusters are much more likely in $\mathrm{CLE}_4$19 than in higher dimensions. It also highlights a major contrast with the 2D continuum GFF sign-cluster theory: in the lattice model the excursion clusters are ordinary connected components of a pointwise excursion set, whereas in the planar continuum model they are non-thin local sets supporting signed Minkowski content measures and restricted Wick powers (Goswami et al., 2021).