(Second-)largest component in
spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche
Papers: tiny.cc/cluster-size-ksrg



Cluster-size decay in supercrit. nearest-neighbor percolation


[Grimmett & Marstrand '90, Kesten & Zhang '90]
Cluster-size decay in supercrit. nearest-neighbor percolation


[Grimmett & Marstrand '90, Kesten & Zhang '90]
Cluster-size decay in Erdős–Rényi random graph








Hyperbolic random graph
Scale-free percolation
Long-range percolation
Scale-free Gilbert RG
Random geom. graph
Nearest-neighbor percolation

Long-range percolation
Vertex set V∞
-
Spatial locations, either
- Lattice Zd
- Poisson point process (unit intensity)

Edge set E∞
- Long-range parameter α>1,
- Edge-density β>0,
- Percolation p∈(0,1]
Connection probability
P(u↔v∣V∞)=p(∥xu−xv∥dβ∧1)α
P(u↔v∣V∞)=p(∥xu−xv∥dβ∧1)α
P(u↔v∣V∞)=p(∥xu−xv∥dβ∧1)α
P(u↔v∣V∞)=p(∥xu−xv∥dβ∧1)α
P(u↔v∣V∞)=p(∥xu−xv∥d1)α∧1
P(u↔v∣V∞)=p(∥xu−xv∥d1∧1)α



Kernel-based spatial random graphs
Vertex set V∞
-
Spatial locations, either
- Lattice Zd
- Poisson point process (unit intensity)
- Power-law i.i.d. weights wv≥1:
P(wv≥w)=w−(τ−1),

Kernel-based spatial random graphs


Edge set E∞
- Long-range parameter α>1,
- Edge-density β>0,
- Percolation p∈(0,1]
Connection probability
P(u↔v∣V∞)=p(∥xu−xv∥dβ⋅(wu⋅wv)∧1)α
Vertex set V∞
-
Spatial locations, either
- Lattice Zd
- Poisson point process (unit intensity)
- Power-law i.i.d. weights wv≥1:
P(wv≥w)=w−(τ−1),
P(u↔v∣V∞)=p(∥xu−xv∥dβ⋅(wu⋅wv)∧1)α
Kernel-based spatial random graphs
Edge set E∞
- Long-range parameter α>1,
- Edge-density β>0,
- Percolation p∈(0,1]


Connection probability
Vertex set V∞
-
Spatial locations, either
- Lattice Zd
- Poisson point process (unit intensity)
- Power-law i.i.d. weights wv≥1:
P(wv≥w)=w−(τ−1),
P(u↔v∣V∞)=p(∥xu−xv∥dβ⋅(wu⋅wv)∧1)α
Components in supercritical graphs
-
Largest component Cn(1):
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations
Questions




Components in supercritical graphs
-
Largest component Cn(1):
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations
Questions




Previous results
[Alexander & Chayes & Chayes '90], [Grimmett & Marstrand, '90], [Kesten & Zhang '90], ..., [Deuschel, Pisztora, '96], [Biskup '04], [Penrose '05], [Sly & Crawford'12], [Kiwi & Mitsche '17], [Lichev, Lodewijks, Mitsche, Schapira '22], [Bläsius, Friedrich, Ruff, Zeiff, '23]



Conjecture: ∃ζ∈[dd−1,1):


Theorem. Long-range percolation (J., Komjáthy, Mitsche '24+)
Set ζ∗:=max{ζlong,(d−1)/d}, with ζlong=2−α.

If d=1 and ζ∗>0.
If d≥2, and
0<ζlong≤(d−1)/d.
ζlong>(d−1)/d.

ζlong≤(d−1)/d, and p or β large.


Conjecture.
Theorem. Geom. Inhom. RG (J., Komjáthy, Mitsche '24+)
Set ζ∗:=max{ζlong,(d−1)/d}, with ζlong=max{2−α,2−(τ−1)/α3−τ}.

If d=1 and ζ∗>0.
If d≥2, and
0<ζlong≤(d−1)/d.
ζlong>(d−1)/d.

ζlong≤(d−1)/d, and p or β large.


Conjecture.
Lower bounds
Upper bounds

Lower bounds: cluster-size decay
Aim: Find minimal ζ s.t.
Aim: Find γ s.t.
Lower bounds: cluster-size decay
Aim: Find minimal ζ s.t.
Aim: Find γ s.t.
Lower bounds: large deviations
(FKG)




Lower bounds: ∣Cn(2)∣
δ small
Upper bounds
Challenge: Delocalized components
# possibilities for ∣2nd-largest∣≥k

Upper bound: ∣Cn(2)∣









Upper bounds

Truncation
Local convergence: giant is almost local
Lower tail of large deviations

mn∼n2−α boxes
Lower tail of large deviations

mn∼n2−α boxes
Upper bounds

What about other value ζ∗?
Description of weight distribution in large components
Prevent "small-to-large" merging
Components in supercritical graphs
- Largest component Cn(1):
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations
Answered questions (d=1)




Tales on tails: large deviations
Lower tail:
Upper tail:
# hubs (wv∼n) required: increase density from ϑ to ϑ+ε


(Second-)largest component in supercritical spatial random graphs
- Cluster-size decay
- Second-largest component
Open problems:
-
Largest component:
- Linear in box size
- Law of large numbers
- Large deviations
Answered questions (d=1)
- Phase transition boundaries.
- Partial results d≥2
- Extension from PPP to grid
- Central limit theorem






- ζ∈[(d−1)/d,1)
Thank you!

Lower bounds
Upper bounds




Small
Large
Lower bounds
Upper bounds
Lower bounds
Upper bounds

Remarks.
- ζ∗<0: subcritical*
-
d≥2:
- (d−1)/d
- partial results: upper bounds
- Product kernel: 2nd term
Power-law degrees: τ>2
d≥2
* [Gracar, Lüchtrath, Mönch '22]
Theorem. (J., Komjáthy, Mitsche '23+)
Set
If ζ∗>0, then LLN for ∣largest∣, and
Long-range parameter: α>1

Example 2:
Geom. Inhomog. RG in d=1
Copy of (Second-)largest component in spatial inhomogeneous random graphs
By joostjor
Copy of (Second-)largest component in spatial inhomogeneous random graphs
- 191