The retention-probability conjecture for Gibbs point processes

Let ${\mathcal S}$ be the state space, let $\lambda(u;\emptyset)$ denote the conditional intensity at $u$ given the empty configuration, and let $\alpha$ be a finite upper bound for the conditional intensity. Let $p_{\lambda,\alpha}(u)$ be the retention probability appearing in the thinning representation, and let $G$ and ${\mathcal V}$ denote the generating functional and void probability, respectively.

Retention-probability conjecture. The function $p_{\lambda,\alpha}(\cdot)$ in the thinning representation is

p_{\lambda,\alpha}(u)=\frac{\lambda(u;\emptyset)}{\alpha}.

Consequently,

G(f)=G(1-g)=\exp\left\{-\int_{\mathcal S}g(u)\lambda(u;\emptyset)\,\mathrm{d}u\right\}

and

{\mathcal V}(B)=\exp\left\{-\int_B\lambda(u;\emptyset)\,\mathrm{d}u\right\}.

The conjecture would give an explicit retention probability and therefore explicit forms for the generating functional and void probability. It is motivated by the expected dependence of the retention probability on the conditional intensity and by the invariance of the representation under the choice of any admissible upper bound $\alpha$ ; the supplied text does not establish the formula.

Progress summary

Solved

An unverified posted counterexample claims the conjecture is false even for a simple hard-core process, while no independent mathematical confirmation has appeared.

Cronie formulated the conjecture in 2024: the thinning retention probability should equal the empty-configuration conditional intensity divided by the stability bound, yielding explicit generating and void probabilities. The paper labels this as unproved and reports only that ongoing work suggests it may hold.

Known results

Last and Otto (2023): a thinning representation with an unspecified retention probability.
Stucki and Schuhmacher (2014): upper and lower bounds for the generating functional of locally stable Gibbs point processes.
Cronie (2024): formulas involving the unknown function $p_{\lambda,\alpha}$ , but no proof of the conjectured identity.

Posted attempt

A reader-written argument claims a counterexample on $S=[0,1]$ with $R>1$ : the normalized hard-core process has void probability $1/2$ , whereas the conjecture predicts $e^{-1}$ . It also identifies an alleged invalid factorization of dependent retention marks. This claimed counterexample has not been independently verified.

Current status (as of August 2026): The conjecture is unproved in the published source, while a posted counterexample claims it is false; neither the refutation nor the original formula has independent verification.

Sources

Sources & referencesView supporting material

Primary source

Ottmar Cronie, “New density/likelihood representations for Gibbs models based on generating functionals of point processes”, arXiv:2406.07075 (2024).

1 week ago

The conjecture is false even for the source's own hard-core Gibbs process. Its asserted generating-functional theorem is also false; in fact, the proposed identity holds exactly for Poisson processes.

Take

S=[0,1],\qquad \beta=\alpha=1,\qquad R>1.

The hard-core Papangelou conditional intensity is

\lambda(u;\xi) = \mathbf1_{\{\xi=\varnothing\}}.

This satisfies the stated local-stability hypothesis

\lambda(u;\xi)\le\alpha=1, \qquad \lambda(u;\varnothing)=1.

Since $R$ exceeds the diameter of $S$ , admissible configurations contain at most one point. Their correctly normalized Janossy densities are

j_0=\frac12,\qquad j_1(u)=\frac12\mathbf1_{[0,1]}(u),\qquad j_r=0\quad(r\ge2).

Therefore the actual void probability and probability generating functional are

\mathcal V(S)=\frac12,\qquad G(f)=\frac12\left(1+\int_0^1f(u)\,du\right).

The conjecture instead predicts

\mathcal V(S) = \exp\left(-\int_0^1\lambda(u;\varnothing)\,du\right) =e^{-1},

and

G(f)= \exp\left(-\int_0^1(1-f(u))\,du\right).

Hence already $f\equiv0$ gives

\boxed{\frac12\ne e^{-1}.}

More generally, every activity $\beta>0$ gives true void probability $1/(1+\beta)$ , while the conjecture predicts $e^{-\beta}$ .

The source's own displayed hard-core Janossy formula also fails normalization: for these parameters it assigns density $e^{-1}$ to both the empty configuration and the singleton stratum, giving total mass

e^{-1}+\int_0^1e^{-1}\,du=\frac2e\ne1.

The obstruction persists even with strictly positive densities and finite interaction energies. For $0<\gamma<1$ , define

Z_\gamma=\sum_{r\ge0}\frac{\gamma^{\binom r2}}{r!}, \qquad j_r(u_1,\ldots,u_r) = \frac{\gamma^{\binom r2}}{Z_\gamma}.

Then

\lambda(u;\xi)=\gamma^{|\xi|}\le1, \qquad \lambda(u;\varnothing)=1,

but

Z_\gamma<e,\qquad \mathcal V_\gamma(S)=Z_\gamma^{-1}>e^{-1}.

Thus a continuum of full-support locally stable Gibbs processes also contradicts the conjecture.

The source's preceding claimed theorem fails because it replaces

\mathbb E\left[ \prod_{x\in Y}(1-M(x)g(x)) \,\middle|\,Y \right]

\prod_{x\in Y} \left(1-\mathbb E[M(x)\mid Y]g(x)\right)

for dependent retention marks. Condition on $Y=\{x,y\}$ , retain exactly one point with equal probabilities, and take $g(x)=g(y)=1$ . The actual expression is $0$ , whereas the incorrectly factorized expression is $1/4$ .

Finally, the sharp corrected characterization is

G(f)= \exp\left[-\int_S(1-f(u))\lambda(u;\varnothing)\,du\right] \text{ for every }f \quad\Longleftrightarrow\quad X\text{ is Poisson with intensity }\lambda(u;\varnothing).

Indeed, the right side is exactly the Poisson generating functional and uniquely determines the point-process law.

Source: arXiv:2406.07075, Conjecture 1, Theorem 1, and the displayed hard-core example.

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Shivam Patel · 2 days ago