Large deviations of the giant in spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche



Junior Probability Seminar University of Oxford, November '24






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

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∞
- Symmetric kernel κ(wu,wv),
- Long-range parameter α∈(1,∞],
- Edge-density β>0,
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
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∞)=(β∥xu−xv∥dκ(wu,wv))α
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
Kernel-based spatial random graphs
Edge set E∞
- Symmetric kernel κ(w1,w2),
- Edge-density β>0,
- Long-range parameter α∈(1,∞],


Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
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),

Geom. Inhom. RG

Hyperbolic RG

Geom. RG

Long-range perc.

Scale-free Gilbert RG

Age-dependent RCM
Theorem. When τ<2 or α<1:
- Infinite degrees
- Bounded diameter
[Deijfen, v.d. Hofstad, Hooghiemstra '13], [Gracar, Grauer, Lüchtrath, Mörters '18]
[Heydenreich, Hulshof, J. '17], [Hirsch '17], [v.d. Hofstad, v.d. Hoorn, Maitra '22],
[J., Komjáthy, Mitsche, '23], [Lüchtrath '22]
Theorem. When τ>2 and α>1:
P(deg(0)≥k)∼k−(τ−1).
τ small: many hubs
α small: many long edges
The interpolating kernel
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
A parameterized kernel: σ≥0
κσ(wu,wv):=max{wu,wv}min{wu,wv}σ
- τ:P(wv≥w)=w−(τ−1).
- σ: assortativity
- σ: interpolation

SFP/GIRG

Hyperbolic RG


Age-dep. RCM
Scale-free Gilbert



Long-range percolation
Random geom. graph
Nearest-neighbor percolation
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]




Lower tail large deviations of the giant*:
One decay exponent governing cluster sizes
*log(log)-corrections at phase transition
Small components*:
If E[#edges of length Ω(n1/d)]→∞
If ζ>(d−1)/d

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
Lower bounds
Upper bounds

Upper bounds
Challenge: Delocalized components
# possibilities for ∣2nd-largest∣≥k

Upper bound: ∣Cn(2)∣









What about other values ζ∗?
κprod=wuwv
κmax=wu∨wv
Description of weight distribution in large components
Prevent "small-to-large" merging
(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!

Supercritical Erdős–Rényi random graph G(n,λ/n)
[Erdős, Rényi '59; O'Connell '98; Andreis, König, Langhammer, Patterson'23]
Exponential growth of neighbourhood
The largest component

Supercritical bond percolation on Zd

[Alexander, Chayes, Chayes '90; Cerf '00; Gandolfi '88; Grimmett & Marstrand '90, Kesten & Zhang '90, Pisztora '96]
Surface-tension driven behavior
The largest component

(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)
Lower bounds
Upper bounds
Lower bounds
Upper bounds
Lower bounds
Upper bounds

Copy of Largest component in spatial random graphs
By joostjor
Copy of Largest component in spatial random graphs
- 111