Gibbs point processes

Question 1

What is a Gibbs point process?

Answer 1

Suppose S ⊆ R^d and Y ∼ po(S, ρ). We say that a point process X has density f : N_{lf}(S) → [0, ∞) with respect to Y if E f (Y) = 1 and for any F ∈ N_{lf},
P(X ∈ F) = E[1{Y ∈ F}f (Y)]
If S is bounded, we say X is a (finite volume) Gibbs point process.

A po(S, ρ)-process has density wrt. po(S, 1)
f_1(x) = exp(|S| − ∫_S ρ(u) du) Π_{u∈x} ρ(u).
If a Gibbs process has density f2 with respect po(S, ρ), then it has density
f (x) = f_1(x)f_2(x)
wrt. a po(S, 1)-distribution.

Question 2

What is a hard-sphere point process?

Answer 2

The hard-sphere process has a density with respect to Y_β ∼ po(S, β)
f(x) = 1/Z Π_{{x,y}⊆X, x≠y} 1{∥x − y∥ > R}
where R>0.

Question 3

What is a Strauss point process?

Answer 3

Let R > 0, β > 0 and 0 ≤ γ ≤ 1. The Strauss process has density
with respect to Y ∼ po(S, 1) given by
f (x) = 1/Z β^{n(x)} Π_{{x,y }⊆x, x≠y} γ^{1{∥x−y ∥≤R}} = 1/Z β^{n(x)} γ^{s_R(x)}
where
s_R(x) = ∑_{{x,y }⊆x, x ≠ y} 1{∥x − y ∥ ≤ R}

Question 4

How is the effect of gamma in the Strauss point process?

Answer 4

γ = 1 is the Poisson process
γ = 0 is the hard-sphere process

Question 5

How is a Strauss point process defined in regards to a Gibbs point process?

Answer 5

The Strauss process is a Gibbs process with pair-wise interactions where ρ(u) = β and ϕ(x, y ) = γ^{1{∥x−y ∥≤R}}.

Question 6

What is the partition function?

Answer 6

Z = E[h(Y)], to normalize the density f.

Question 7

What is Ruelle stability?

Answer 7

There is a constant c > 0 and a function α : S → R such that ∫_S α(x) d x < ∞ and h({x_1, . . . , x_n}) ≤ c Π_{i=1}^n α(x_i)

If h is Ruelle stable, then Z < ∞

Question 8

What is hereditary?

Answer 8

For h : N_f (S) → [0, ∞):
h(x) = 0 implies h(x ∪ {u}) = 0 for all u ∈ S.

Question 9

What is non-degenerate?

Answer 9

For h : N_f (S) → [0, ∞):
h(∅) > 0

Question 10

What is the Papangelou conditional intensity?

Answer 10

Suppose X has a hereditary density f proportional to h with respect to a Y. The Papangelou conditional intensity (PI) is given for x ∈ N_f and u ∈/ x by
λ(x, u) = f (x ∪ {u}) / f (x) = h(x ∪ {u}) / h(x) ,
where we interpret 0/0 as 0.

Question 11

What is the interpretation of the Papangelou conditional intensity?

Answer 11

Consider a small set B around a point u with volume |B| = d u. Suppose X_{S\B} = y. The probability of finding a point in B is ≈λ(y, u) du.

Question 12

What is a attractive/repulsive point process?

Answer 12

We say that the process is
Attractive: For x ⊆ y, λ(x, u) ≤ λ(y, u).
Repulsive: For x ⊆ y, λ(x, u) ≥ λ(y, u).

Question 13

How is the GNZ formula defined and what does it generalize?

Answer 13

Assume X is a Gibbs process on S with hereditary and Ruelle stable density proportional to h with respect to Y ∼ po(S, 1). Let g : S × Nf (S) → [0, ∞). Then

E[ ∑_{u∈X} g (u, X{u})] = E[ ∫_S g(u, X)λ(X, u) du}].

Generalizes Slivnyak-Mecke for the Poisson case

Question 14

What are the DLR equations?

Answer 14

Suppose that a Gibbs process X has density f on S. Let B ⊆ S. For almost all X_{S\B}, the conditional distribution of X_B given X_{S\B} = y is a Gibbs process on B with density
f (x | y) = f (x ∪ y) / f_{S\B} (y) = f (x ∪ y) / E[f (YB ∪ y)],
x ∈ N_f (B), with respect to Y_B ∼ po(B, 1).

Question 15

What is a finite interaction range?

Answer 15

Let h : N_f → [0, ∞) be a Ruelle stable and hereditary function. We say that h has finite interaction range R if for all u ∈ R^d and x ∈ N_f,
λ(x, u) = λ(x ∩ b(u, R), u)

The Strauss process has finite interaction range