In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, physics, and wireless network modeling.
Let
Λ
be a locally compact Polish space and
μ
be a Radon measure on
Λ
. Also, consider a measurable function K:Λ^{2} → ℂ.
We say that
X
is a determinantal point process on
Λ
with kernel
K
if it is a simple point process on
Λ
with a joint intensity or correlation function (which is the density of its factorial moment measure) given by
ρ
n
(
x
1
,
…
,
x
n
)
=
det
(
K
(
x
i
,
x
j
)
1
≤
i
,
j
≤
n
)
for every n ≥ 1 and x_{1}, . . . , x_{n} ∈ Λ.
The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρ_{k}.
Symmetry: ρ_{k} is invariant under action of the symmetric group S_{k}. Thus:
Positivity: For any N, and any collection of measurable, bounded functions φ_{k}:Λ^{k} → ℝ, k = 1,. . . ,N with compact support:
If
Then
A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρ_{k} is
∑
k
=
0
∞
(
1
k
!
∫
A
k
ρ
k
(
x
1
,
…
,
x
k
)
d
x
1
⋯
d
x
k
)
−
1
k
=
∞
for every bounded Borel A ⊆ Λ.
The eigenvalues of a random m × m Hermitian matrix drawn from the Gaussian unitary ensemble (GUE) form a determinantal point process on
R
with kernel
K
m
(
x
,
y
)
=
∑
k
=
0
m
−
1
ψ
k
(
x
)
ψ
k
(
y
)
where
ψ
k
(
x
)
is the
k
th oscillator wave function defined by
ψ
k
(
x
)
=
1
2
n
n
!
H
k
(
x
)
e
−
x
2
/
4
and
H
k
(
x
)
is the
k
th Hermite polynomial.
The poissonized Plancherel measure on partitions of integers (and therefore on Young diagrams) plays an important role in the study of the longest increasing subsequence of a random permutation. The point process corresponding to a random Young diagram, expressed in modified Frobenius coordinates, is a determinantal point process on ℤ + ^{1}⁄_{2} with the discrete Bessel kernel, given by:
K
(
x
,
y
)
=
{
θ
k
+
(

x

,

y

)

x

−

y

if
x
y
>
0
,
θ
k
−
(

x

,

y

)
x
−
y
if
x
y
<
0
,
where
k
+
(
x
,
y
)
=
J
x
−
1
2
(
2
θ
)
J
y
+
1
2
(
2
θ
)
−
J
x
+
1
2
(
2
θ
)
J
y
−
1
2
(
2
θ
)
,
k
−
(
x
,
y
)
=
J
x
−
1
2
(
2
θ
)
J
y
−
1
2
(
2
θ
)
+
J
x
+
1
2
(
2
θ
)
J
y
+
1
2
(
2
θ
)
For J the Bessel function of the first kind, and θ the mean used in poissonization.
This serves as an example of a welldefined determinantal point process with nonHermitian kernel (although its restriction to the positive and negative semiaxis is Hermitian).
Let G be a finite, undirected, connected graph, with edge set E. Define I^{e}:E → ℓ^{2}(E) as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edge e, define I^{e} to be the projection of a unit flow along e onto the subspace of ℓ^{2}(E) spanned by star flows. Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel
K
(
e
,
f
)
=
⟨
I
e
,
I
f
⟩
,
e
,
f
∈
E
.