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Lecture 13: Gibbs Point Processes (+ Intro to DPPs)

Introduction

Last time: we defined Janossy measures for a finite point process
$ = {i=1}^{N} {X_i} $ on $E \subseteq \mathbb{R}^d$.

Definition (Janossy Measures). For $n\in\mathbb{N}$, the $n$-th Janossy measure of $\Phi$ is the measure $J_n$ on $E^n$ defined by \[ J_n(B_1 \times \cdots \times B_n) = n! \, P_n \, \Pi_n^{\mathrm{sym}}(B_1 \times \cdots \times B_n), \qquad B_i \in \mathcal{B}(E), \] where $P_n = \mathbb{P}[N=n]$ and $\Pi_n^{\mathrm{sym}}$ is the joint distribution of $(X_1,\ldots,X_n)$ conditioned on $N=n$.

Remarks:

$J_n$ is sometimes defined without the $n!$ constant. In that case one would have
$J_n(E^n) = \mathbb{P}[N=n]\Pi_n^{\mathrm{sym}}(B_1 \times \cdots \times B_n)$ instead of $J_n(E^n) = n! \, \mathbb{P}[N=n]\Pi_n^{\mathrm{sym}}(B_1 \times \cdots \times B_n)$.
Janossy measures (unlike moment measures and product densities) depend strongly on the observation region.
If we have a point process on $\mathbb{R}^d$, we may define its local Janossy measures and densities for any bounded $E \subset \mathbb{R}^d$ by restricting to $E$.
If local Janossy measures are given, one can define moment measures and finite–dimensional distributions.
Existence of point processes from finite–dimensional distributions can be shown using Kolmogorov consistency conditions (Theorem 9.2.X in [daley2008introduction]).

Gibbs Point Processes

Assume $E \subset \mathbb{R}^d$ is compact.
Let $\Phi$ be a homogeneous Poisson process on $E$ with unit intensity. Let $f : \mathcal{M}(E) \to \mathbb{R}_+$ be measurable such that $ = 1 . $

Define the probability measure $\mathbb{P}_f$ on $\mathcal{M}(E)$ by, for $A\in\mathscr{M}(E)$, \[ \mathbb{P}_f(A) := \mathbb{E}\big[\, f(\Phi)\, \mathbf{1}_{\{\Phi\in A\}} \big]. \]

Definition. A point process $\Phi_f$ with distribution $\mathbb{P}_f$ is said to have
probability density $f$ (with respect to the unit–intensity Poisson process).

Definition. A Gibbs point process on $E$ is a point process whose density is of the form \[ f\!\left( \sum_{i=1}^{n} \delta_{x_i} \right) = \alpha \exp\!\big[\, \mathcal{U}(x_1,\ldots,x_n) \big], \] where \[ \mathcal{U}(x_1,\ldots,x_n) = \sum_{k=1}^{n} \; \sum_{\{i_1,\ldots,i_k\}\subset \{1,\ldots,n\}} V_k(x_{i_1},\ldots,x_{i_k}), \] and $V_k(\cdot)$ is the potential of order $k$.
The constant $\alpha$ is a normalizing constant ensuring $ = 1. $

Janossy Densities for the Homogeneous Poisson Process

For a homogeneous Poisson process on $E$ with intensity $1$, the Janossy densities satisfy \[ j_n(x_1,\ldots,x_n) = e^{-|E|}, \] where $|E|$ denotes the Lebesgue measure of $E$.

Thus for any measurable $g:\mathcal{M}(E)\to\mathbb{R}_+$,

\[\begin{align*} \mathbb{E}\big[g(\Phi_f)\big] &= \mathbb{E}\big[g(\Phi)\, f(\Phi)\big] \\ &= e^{-|E|} \sum_{n=0}^{\infty} \frac{1}{n!} \int_{E^n} g\!\left(\sum_{i=1}^{n} \delta_{x_i}\right) f\!\left(\sum_{i=1}^{n} \delta_{x_i}\right) \, dx_1 \cdots dx_n . \end{align*}\]

Comparing with the definition of Janossy measures, we see that Gibbs point processes on $E$ can be defined as point processes with Janossy densities \[ j_n(x_1,\ldots,x_n) \;\propto\; \exp\!\big[\, \mathcal{U}(x_1,\ldots,x_n) \big], \] where $\mathcal{U}$ is as above.

Examples

Example (Strauss Process). Let $\beta>0$, $0 \le \gamma \le 1$, and $r>0$.

Define the probability density
$$
f\!\left( \sum_{i=1}^{n} \delta_{x_i} \right)
= \alpha \, \beta^{\, n} \, \gamma^{\, S(x_1,\ldots,x_n)},
$$
where
$$
S(x_1,\ldots,x_n)
= \sum_{i<j} \mathbf{1}_{\{\|x_i - x_j\|\le r\}} .
$$

Then the Gibbs point process $\Phi_f$ with density $f$ is called a *Strauss process*.

If $\gamma = 1$, this is a Poisson process of intensity $\beta$.
If $\gamma = 0$, this is a hard–core process.
For $0 < \gamma < 1$, the interaction is inhibitory (repulsive).

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