Weak Gibbs Measures in Dynamical Systems

• Weak Gibbs measures are Gibbs-type objects that relax the strict uniform estimates of classical Gibbs measures by allowing subexponential errors while preserving exponential asymptotics.
• They are utilized in thermodynamic formalism, symbolic dynamics, and fractal geometry to study equilibrium states, multifractal spectra, and large deviation principles.
• Their flexible framework supports analysis in non-Hölder or intermittent systems and facilitates understanding of local dependencies and random dynamics in advanced applications.

Weak Gibbs measures are Gibbs-type objects in thermodynamic formalism, multifractal analysis, fractal geometry, and generalized Gibbsianity obtained by relaxing the uniform finite-scale estimates of classical Gibbs measures while preserving the correct exponential asymptotics. In symbolic and smooth dynamical settings, the relaxation is usually a subexponential distortion in word length or time; in generalized Gibbs theory it can instead mean that an interaction potential converges only on a full-measure set. The terminology is therefore not uniform across subfields, but a common theme is that the logarithmic scaling of cylinder sets, Bowen balls, or local partition elements remains controlled even when the classical bounded-distortion constants are lost (Iommi et al., 2015, Pfister et al., 2017, Kesseböhmer et al., 2021, Ny, 2013).

1. Terminology and principal definitions

The classical Gibbs property on a symbolic system with potential $\phi$ is the uniform estimate

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$

with one constant $C$ independent of $n$ and of the cylinder. Weak Gibbs theory replaces this by a subexponential error. In the asymptotically additive setting of a topologically mixing Markov shift, a measure $\mu$ is weak Gibbs for $\Phi=(\phi_n)_n$ if there exists $K(n)>0$ with

$\lim_{n\to\infty}\frac{\log K(n)}{n}=0$

such that

$$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$

for every $n$, cylinder $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$0, and $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$1 (Iommi et al., 2015).

For compact dynamical systems $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$2, an alternative but equivalent exponential-scale formulation is given in terms of Bowen balls. A probability measure $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$3 is weak Gibbs for $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$4 if for every $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$5 there exists $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$6 such that for all sufficiently large $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$7,

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$8

uniformly in $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$9 (Pfister et al., 2017). This is a Bowen-ball version of the same idea: $C$0 is asymptotic to the Birkhoff sum $C$1 up to $C$2.

On self-conformal attractors generated by a $C$3-IFS $C$4, weak Gibbs measures arise from a continuous potential $C$5 satisfying $C$6 and a dual fixed point $C$7 of $C$8. The pushforward $C$9 is a weak $n$0-Gibbs measure, and its cylinder masses satisfy

$n$1

where $n$2. The distortion is thus subexponential rather than uniformly bounded (Kesseböhmer et al., 2021).

On countable Markov shifts, the paper on local weak $n$3-Gibbs measures introduces an additional local dependence on the initial symbol. A probability $n$4 is a local weak Gibbs measure for $n$5 if

$n$6

with $n$7 depending on the first symbol and $n$8 (Pérez, 2016).

In generalized Gibbsianity, the phrase has a different formal content. A measure $n$9 is weakly Gibbs if there exists a potential $\mu$0 and a tail-measurable set $\mu$1 with $\mu$2 such that $\mu$3 is absolutely convergent on $\mu$4 and $\mu$5 for some $\mu$6 (Ny, 2013).

Setting	Representative estimate	Weak feature
Compact $\mu$7	$\mu$8	asymptotic Bowen-ball control
Mixing Markov shift	$\mu$9	$\Phi=(\phi_n)_n$0
Self-conformal IFS	cylinder mass comparable to $\Phi=(\phi_n)_n$1	$\Phi=(\phi_n)_n$2
Countable shift	same, with factor $\Phi=(\phi_n)_n$3	local dependence on first symbol
Generalized Gibbsianity	potential converges on $\Phi=(\phi_n)_n$4 only	almost-sure absolute convergence

2. Thermodynamic formalism and equilibrium structure

Weak Gibbs measures retain much of the equilibrium-state structure of classical Gibbs measures. In the asymptotically additive framework, every weak Gibbs measure is an exact Gibbs measure for another asymptotically additive sequence. If $\Phi=(\phi_n)_n$5 is weak Gibbs for $\Phi=(\phi_n)_n$6, the sequence

$\Phi=(\phi_n)_n$7

is asymptotically additive, satisfies $\Phi=(\phi_n)_n$8, and yields the exact identity

$\Phi=(\phi_n)_n$9

for $K(n)>0$0 (Iommi et al., 2015). In that sense, weak Gibbsness is not merely a defective form of Gibbsianity; it becomes exact after enlarging the class of potentials.

For weak Gibbs measures on compact dynamical systems, the thermodynamic normalization is built into the definition. If $K(n)>0$1 is weak Gibbs for $K(n)>0$2, then

$K(n)>0$3

If in addition $K(n)>0$4, then

$K(n)>0$5

so $K(n)>0$6 is an equilibrium measure for $K(n)>0$7 (Pfister et al., 2017). The weak Gibbs property thus identifies the correct exponential scaling and simultaneously fixes the pressure normalization.

A converse direction holds in broad symbolic classes. For a shift space $K(n)>0$8 satisfying the decoupling condition, every tangent functional to the pressure at $K(n)>0$9, equivalently every equilibrium measure for an absolutely summable potential $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$0, is a weak Gibbs measure for

$\lim_{n\to\infty}\frac{\log K(n)}{n}=0$1

In dimension $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$2, the same conclusion holds under the weaker 1-decoupling condition, and irreducible sofic shifts satisfy that condition (Pfister et al., 2019). This extends one direction of the classical equilibrium/Gibbs correspondence to subshifts without full specification.

A sharp criterion of the same type appears for natural extensions of $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$3-shifts. For $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$4, $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$5, and $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$6 with bounded total oscillations, an equilibrium measure is weak Gibbs if and only if

$\lim_{n\to\infty}\frac{\log K(n)}{n}=0$7

while weak Gibbs fails when

$\lim_{n\to\infty}\frac{\log K(n)}{n}=0$8

(Yamashita, 30 Sep 2025). Here the obstruction is combinatorial and is encoded in the growth of the distinguished-prefix quantity $\lim_{n\to\infty}\frac{\log K(n)}{n}=0$9.

For a broad class of local homeomorphisms satisfying a positive-frequency good-times condition, a Bowen property on good dynamical balls, and a pressure gap

$$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$0

there exists a unique ergodic weak Gibbs measure $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$1. If $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$2 admits a generating partition, that measure is also the unique equilibrium state (Ferreira et al., 23 Oct 2025). In this setting the Gibbs estimate holds at a sequence of Gibbs times rather than at all times.

3. Countable-state, random, and non-uniform extensions

Weak Gibbs theory is particularly useful when uniform symbolic control is unavailable. On topologically mixing countable Markov shifts, the local weak $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$3-Gibbs formalism accommodates both subexponential distortion and dependence on the initial symbol. This permits a Bowen-type dimension theory under BI and BIP hypotheses, including systems for which fully uniform Gibbs bounds are unavailable (Pérez, 2016). The point of the “local” modifier is precisely that in the countable-state setting one often cannot remove the $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$4-dependence.

Random weak Gibbs measures extend the same principle to quenched random dynamics. In the random subshift framework associated with a random $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$5 system, one has random eigenmeasures $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$6 and projected measures $$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$7 satisfying

$$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$8

together with geometric estimates

$$\frac{1}{K(n)} \le \frac{\mu(C_{i_1\cdots i_n})}{\exp(\phi_n(\omega)-nP(\Phi))} \le K(n)$$9

The potentials need only be continuous along fibers, and the resulting theory applies to random weak Gibbs measures on attractors generated by $n$0 random dynamics semiconjugate to random subshifts of finite type (Yuan, 2016).

A related construction appears for inverse measures of random weak Gibbs measures. If $n$1 has zero Lebesgue measure, the inverse measure $n$2 of $n$3 is discrete and can be written explicitly as a weighted sum of Dirac masses located at distribution-function images of endpoint data. This discrete structure is central to the inverse multifractal formalism developed for $n$4 (Yuan, 2017).

The local-homeomorphism theory provides a different non-uniform extension. There the weak Gibbs property takes the form

$n$5

along a point-dependent sequence of Gibbs times $n$6. If the Gibbs times are non-lacunar, the estimates between consecutive Gibbs times acquire subexponential corrections of the form

$n$7

which again places weak Gibbsness at the level of exponential asymptotics rather than exact finite-time distortion bounds (Ferreira et al., 23 Oct 2025).

4. Multifractal, dimensional, and large-deviation consequences

Weak Gibbs measures are strong enough to support a substantial multifractal and large-deviation theory. For compact dynamical systems, the Bowen-ball formulation immediately yields large deviation bounds for empirical measures. If $n$8 is weak Gibbs for $n$9, then for open $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$00 and ergodic $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$01,

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$02

while for closed convex $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$03,

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$04

(Pfister et al., 2017). Under upper semicontinuity of entropy and entropy density of ergodic measures, these bounds become a full large deviation principle.

In shrinking-target theory, local weak $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$05-Gibbs measures furnish generalized Bowen formulas. For target sets

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$06

the $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$07-dimension is bounded above and below by pressure expressions involving $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$08, and in finite-alphabet cases one gets a precise Bowen equation

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$09

for the dimension (Pérez, 2016). Because only weak Gibbs control is required, the theory covers non-Hölder potentials and intermittent systems, including the Manneville–Pomeau map.

Random weak Gibbs measures also satisfy the full multifractal formalism. For $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$10-a.e. $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$11, the $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$12-spectrum of $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$13 is

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$14

where $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$15 is defined by

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$16

and the exact Hausdorff spectrum is

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$17

for $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$18 (Yuan, 2016). The same paper computes Hausdorff and packing dimensions of divergent local-dimension sets and proves $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$19-$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$20 laws for Hausdorff and packing measures.

For inverse measures of random weak Gibbs measures, the spectrum changes in a characteristic way: $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$21 On the principal interval one has

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$22

while for the lower spectrum there is a linear branch

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$23

on $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$24, reflecting the contribution of the gaps of the Cantor attractor and the atomic nature of the inverse measure (Yuan, 2017).

The local-homeomorphism framework likewise yields a large deviations principle for the unique weak Gibbs measure, with upper bounds involving both the variational term

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$25

and an error term controlling the sparsity of Gibbs times, and lower bounds over ergodic invariant measures supported on the good set $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$26 (Ferreira et al., 23 Oct 2025).

5. Self-conformal fractals, overlaps, and spectral theory

Weak Gibbs measures play a central role in the spectral theory of one-dimensional Krein–Feller operators on self-conformal sets. For a non-trivial $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$27-IFS on $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$28, weak $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$29-Gibbs measures provide enough control to treat both nonlinear conformal systems and systems with overlaps. The decisive cylinder estimate is subexponential, not uniform, and this is exactly what allows the theory to survive beyond the classical Gibbs setting (Kesseböhmer et al., 2021).

The multifractal quantity governing the spectral problem is the $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$30-spectrum

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$31

For every weak Gibbs measure on the unit interval with respect to a non-trivial $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$32-IFS, $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$33 exists as a true limit on $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$34. This limit property is significant because it holds with or without overlaps (Kesseböhmer et al., 2021).

The main spectral consequence is that the spectral dimension $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$35 exists and equals the fixed point of the $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$36-spectrum: $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$37 Equivalently, if $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$38 is defined by $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$39 and $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$40, then

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$41

for weak Gibbs measures associated with a non-trivial $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$42-IFS, again with or without overlaps (Kesseböhmer et al., 2021).

Under the open set condition, the fixed-point description becomes a pressure formula. With

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$43

the spectral dimension is the unique zero $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$44 of $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$45: $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$46 In the self-similar case this recovers the classical equation

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$47

Under stronger assumptions—$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$48 Hölder, the IFS $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$49, and OSC—the eigenvalue counting function satisfies the two-sided asymptotic law

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$50

The theorem gives comparability rather than a precise asymptotic equivalent. A further corollary states that if $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$51, then $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$52 is singular with respect to Lebesgue measure (Kesseböhmer et al., 2021).

6. Generalized Gibbsianity and terminological cautions

In lattice statistical mechanics, weak Gibbs measures belong to the broader theory of generalized Gibbs measures and must be sharply distinguished from weak limits of Gibbs states. The canonical example is the decimation of the low-temperature two-dimensional Ising model. The decimated measure is not quasilocal, hence not Gibbs in the usual DLR-plus-quasilocal sense, because the alternating configuration produces a hidden phase transition on the constrained decorated lattice. Nevertheless, the decimated measure is weakly Gibbs: there exists a translation-invariant potential absolutely convergent on a full-measure set and consistent with the decimated specification (Ny, 2013).

The same decimated measure is also almost Gibbs. The paper explicitly records the hierarchy

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$53

and proves that the decimated extremal phases are both almost Gibbs and weakly Gibbs (Ny, 2013). Beyond mere almost-sure absolute convergence, the weakly Gibbsian potential can be refined to satisfy a quenched correlation decay estimate: for a random length $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$54,

$$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$55

when $$C^{-1}\le \frac{\mu(C_n(\omega))}{\exp(S_n\phi(\omega)-nP(\phi))}\le C,$$56. This gives a configuration-dependent effective range rather than a uniform summability bound.

Several neighboring literatures use the adjective “weak” differently and should not be conflated with weak Gibbs measures. In low-temperature Ising and Potts theory, one studies Gibbs states that are not weak limits of finite-volume Gibbs measures with deterministic boundary conditions; there “weak” refers to weak convergence, not weak Gibbsianity (Coquille, 2014). In loopy belief propagation, convergence is characterized by the existence of a weak limit of Gibbs measures on computation trees, again in the topological sense of weak convergence (Tatikonda et al., 2012). On trees and free groups, Glauber dynamics may converge weakly to the set of ordinary Gibbs measures, but the paper explicitly studies standard DLR Gibbs measures rather than a weaker Gibbs class (Shriver, 2020). In planar FK-percolation, the specification has a weaker spatial Markov property, but the theory still concerns standard Gibbs measures for a nonlocal specification, not weak Gibbs measures (Glazman et al., 2021).

For this reason, “weak Gibbs measure” is best understood as a family resemblance term rather than a single definition. In symbolic dynamics, fractal geometry, and random thermodynamic formalism it denotes Gibbs asymptotics with subexponential distortion; in generalized Gibbsianity it denotes almost-sure convergence of an interaction potential on a full-measure set; and in adjacent areas one must verify whether “weak” refers instead to topology, coupling strength, or a weakened spatial Markov property.