Large deviations of the giant in spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche
Long-range phenomena in percolation, September '24
Supercritical bond percolation on Zd
P(k<∣C(0)∣<∞)∼
\mathbb{P}\big(k< |{\color{green}\mathcal{C}(0)}|<\infty\big)\sim
exp(−Θ(kζ))
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big)
[Gandolfi '88, Grimmett & Marstrand '90, Kesten & Zhang '90, Pisztora '96]
∣Cn(2)∣∼P(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}|\sim_\mathbb{P}(\log n)^{1/\zeta}
Surface-tension driven behavior
P(∣Cn(1)∣/n<θ−ε)∼
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)\sim
exp(−Θ(nζ))
\exp\big(-\Theta(n^{\mathbf{\zeta}})\big)
θ(p):=P(0↔∞)>0
\theta(p):=\mathbb{P}(0\leftrightarrow \infty)>0
ζ=dd−1
\zeta=\frac{d-1}d
The largest component
P(∣Cn(1)∣/n>θ+ε)∼
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)\sim
exp(−Θ(n))
\exp\big(-\Theta(n)\big)
Soft Poisson Boolean model
Edge set E∞
- {u↔v}⟺{wu1/d+wv1/d≥∥xu−xv∥}
- Long-range parameter α>1,
- Percolation p∈(0,1]
Connection probability
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
Vertex set V∞
- Locations: Poisson point process intensity β>0
- Power-law i.i.d. weights τ>2:
P(wv≥w)=w−(τ−1), w≥1
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
(deg(v)∣wv=w)=Poi(cα,d,τpβ⋅w)
Soft Poisson Boolean model
Supercriticality assumption
- θ=θd,τ,α,p(β):=Pd,τ,α,p,β0(0↔∞)
- β>βc(α,τ,d,p)=inf{β:θ(β)>0}
Edge set E∞
- {u↔v}⟺{wu1/d+wv1/d≥∥xu−xv∥}
- Long-range parameter α>1,
- Percolation p∈(0,1]
Vertex set V∞
- Locations: Poisson point process intensity β>0
- Power-law i.i.d. weights τ>2:
P(wv≥w)=w−(τ−1), w≥1
Soft Poisson Boolean model
Questions:
Fixed parameters:
- Point process on Rd intensity β>0
- Power-law exponent τ>2
- Long-range parameter α>1
- Percolation p∈(0,1]
[Gracar, Lüchtrath, Mönch '21]
- When is βc<∞?
- d≥2,
- Size of the largest connected component ∣Cn(1)∣
- Law of large numbers (∣Cn(1)∣/n→θ)?
- surface-tension behav. in P(∣Cn(1)∣/n<θ−ε)?
- exponential decay of P(∣Cn(1)∣/n>θ+ε)?
or d=1, and α∈(1,2∨ατ)
Power-law weights
Constant weights
Lower tail*:
If E[#edges of length n1/d]→∞
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α)∨(1−αα−1(τ−1)))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee(1-\frac{\alpha-1}\alpha(\tau-1))}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
Main result: large deviations for the giant component
*log-corrections at phase transition
Upper tail:
If non-critical
P(∣Cn(1)∣/n>θ+ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
∼∗n−I(θ+ε)
\sim^\ast n^{-I(\theta+\varepsilon)}
P(∣Cn(1)∣/n>θ+ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
*at continuity points of I(θ+ε)
⋙
\ggg
⋘
\lll
∼exp(−n)
\sim \exp(-n)
Biskup ['04]
P(∣Cn(1)∣/n<ε)
\mathbb{P}(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \varepsilon)
≲exp(−n2−α−o(1))
\lesssim\exp\big(-n^{2-\alpha-o(1)}\big)
Lower tail: a too small largest component
∣largest∣/n⟶Pθ:=P0(0↔∞)
|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \theta:=\mathbb{P}^0\big(0\leftrightarrow\infty\big)
P(∣largest∣/n<ϑ−ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\vartheta- \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
Goal 1:
P0(∣Cn(0)∣>k,0∈/Cn(1))
\mathbb{P}^0(|\mathcal{C}_n(0)|>k, 0\notin\mathcal{C}_n^{(1)})
⟶k→∞0
\overset{k\to\infty}\longrightarrow 0
+ local convergence (Giant is almost local)
[vdHofstad, vdHoorn, Maitra '21, vd Hofstad '21]
Goal 2: Prove the theorem
Bootstrapping: Goal 1 proved via method Goal 2
Edge set E∞
- Long-range parameter α>1,
- Percolation p∈(0,1]
Connection probability
Vertex set V∞
- Locations: Poisson point process intensity β>0
- Power-law i.i.d. weights τ>2:
P(wv≥w)=w−(τ−1), w≥1
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
P(∣largest∣/n<ϑ−ε)≲exp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\vartheta- \varepsilon \big)\lesssim \exp\big(-n^\zeta\big)
Goal 2 Prove
∣largest∣/n⟶Pθ
|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \theta
⟹
\Longrightarrow
Power-law weights
Constant weights
Lower tail*:
If E[#edges of length n1/d]→∞
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α)∨(1−αα−1(τ−1)))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee(1-\frac{\alpha-1}\alpha(\tau-1))}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
Main result: large deviations for the giant component
*log-corrections at phase transition
Upper tail:
If non-critical
P(∣Cn(1)∣/n>θ+ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
∼∗n−I(θ+ε)
\sim^\ast n^{-I(\theta+\varepsilon)}
P(∣Cn(1)∣/n>θ+ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
*at continuity points of I(θ+ε)
⋙
\ggg
⋘
\lll
∼exp(−n)
\sim \exp(-n)
Upper tail: polynomial decay when τ<∞
P(∣Cn(1)∣/n>ρ)=
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)=
Related work:
- Sum of iid Par(τ): principle of single jump.
- Large deviations # edges En (vdHvdHKMM '24)
P(En>E[En]+nν)=∗(1+o(1))cνn−(τ−2)⌈ν⌉. - Inhomogeneous random graphs (J., Zwart '24+)
P(∣Cn(1)∣/n>ρ)=∗(1+o(1))Cρn−(τ−2)⌈h(ρ)⌉.
Θ(n−(τ−2)⌈h(ρ)⌉)
\Theta(n^{-(\tau-2)\lceil h(\rho)\rceil})
P(u↔v∣V∞)=p(∥xu−xv∥dwu1/d+wv1/d∧1)dα
Challenge: Capture dependency on percolation p
Geom. inhom. RG
Long-range percolation
Lower tail*:
If E[#edges of length n1/d]→∞
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α)∨2−(τ−1)/α3−τ)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee\frac{3-\tau}{2-(\tau-1)/\alpha}}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
Main result: large deviations for the giant component
*log-corrections at phase transition
Upper tail:
If non-critical, ρ∈(θ,1)
P(∣Cn(1)∣/n>ρ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
∼∗n−I(ρ)
\sim^\ast n^{-I(\rho)}
P(∣Cn(1)∣/n>ρ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
*at continuity points of I(ρ)
⋙
\ggg
⋘
\lll
∼exp(−n)
\sim \exp(-n)
Thank you!
-
Cluster-size decay in supercritical long-range percolation;
With Júlia Komjáthy, Dieter Mitsche;
Electronic Journal of Probability (2024). -
Cluster-size decay in supercritical KSRGs;
With Júlia Komjáthy, Dieter Mitsche;
Preprint arXiv: 2303.00712. -
Large deviations of the giant in supercritical KSRGs;
With Júlia Komjáthy, Dieter Mitsche;
Preprint arXiv: 2404.02984.
Kernel-based spatial random graphs
Vertex set V∞
- Spatial locations xv∈Rd: PPP(1),
- i.i.d. weights wv≥1: Pareto(τ),
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 xv∈Rd: PPP(1), or Zd,
- i.i.d. weights wv≥1: P(wv≥w)=w−(τ−1),
Edge set E∞
- Symmetric kernel κ(w1,w2),
- Edge-density β>0,
- Long-range parameter α>1,
κprod=wuwv
\kappa_\mathrm{prod}=w_uw_v
α∈(1,∞],d≥1
\alpha\in(1,\infty], d\ge 1
Geom. Inhom. RG
κprod=wuwv
\kappa_\mathrm{prod}=w_uw_v
α=∞,d=1
\alpha=\infty, d= 1
Hyperbolic RG
Geom. RG
κtriv≡1
\kappa_\mathrm{triv}\equiv 1
α=∞,d≥2
\alpha=\infty, d \ge 2
Long-range perc.
κtriv≡1
\kappa_\mathrm{triv}\equiv 1
α∈(1,∞),d≥1
\alpha\in(1,\infty), d\ge 1
κmax=wu∨wv
\kappa_{\max}=w_u\vee w_v
Scale-free Gilbert RG
κpa=(wu∨wv)
\kappa_{\mathrm{pa}}=(w_u\vee w_v)
⋅(wu∧wv)τ−2
\cdot(w_u\wedge w_v)^{\tau-2}
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
τ−1σ
\frac{\sigma}{\tau-1}
τ−11
\frac1{\tau-1}
1
1
1
1
σ=τ−1
\sigma=\tau-1
τ=2
\tau=2
σ=1
\sigma=1
σ=1
\sigma=1
σ=τ−2
\sigma=\tau-2
σ=τ−2
\sigma=\tau-2
σ=0
\sigma=0
σ=0
\sigma=0
SFP/GIRG
Hyperbolic RG
Age-dep. RCM
Scale-free Gilbert
Long-range percolation
Random geom. graph
Nearest-neighbor percolation
Remarks.
- ζ∗<0: subcritical*
-
d≥2:
- (d−1)/d
- partial results: upper bounds
- Product kernel: 2nd term
No known results
- ∣largest∣/n
- 2nd-largest
Example 1:
Scale-free Gilbert RG in d=1
Power-law degrees: τ>2
d≥2
ζ∗:=max{2−α,(3−τ)/(2−(τ−1)/α)}.
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(3\!-\tau)/(2\!-\!(\tau\!-\!1)/\alpha)
\}.
\end{aligned}
* [Gracar, Lüchtrath, Mönch '22]
Theorem. (J., Komjáthy, Mitsche '23+)
Set
ζ∗:=max{2−α,(τ−2)−(τ−1)/α}.
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(\tau\!-\!2)\!-\!(\tau\!-\!1)/\alpha
\}.
\end{aligned}
If ζ∗>0, then LLN for ∣largest∣, and
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\color{blue}\text{largest}}|\big]}\big)
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∣2nd-largest∣∼(logn)1/ζ∗
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Long-range parameter: α>1
κmax=wu∨wv
κprod=wuwv
Example 2:
Geom. Inhomog. RG in d=1
Lower bounds: cluster-size decay
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≳exp(−kζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\le |\mathcal{C}(0)|<\infty\big)
∼P(Qk isolated)
\sim\mathbb{P}\big(\mathcal{Q}_k\text{ isolated}\big)
∼exp(−kζ)
\sim \exp\big(-k^\zeta\big)
∣C(0)∣≥k
|\mathcal{C}(0)|\ge k
}
\Bigg\}
Θ(k)
\Theta(k)
E[∣□□∣]∼E[∣□□↔□□∣]
\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]
Aim: Find γ s.t.
∼kζ
\sim k^\zeta
Lower bounds: cluster-size decay
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≳exp(−kζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);
∣C(0)∣≥k
|\mathcal{C}(0)|\ge k
}
\Bigg\}
Θ(k)
\Theta(k)
kζ∼E[∣{v∈Λk:v↘Λkc}∣]
k^\zeta\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\searrow \Lambda_k^c\}|\big]
E[∣□□∣]∼E[∣□□↔□□∣]
\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]
Aim: Find γ s.t.
∼kζ
\sim k^\zeta
Lower bounds: large deviations
0
0
n
n
}
\Bigg\}
ρn/2
\rho n/2
P(∣Cn(1)∣≤ρn)
\mathbb{P}\big(|\mathcal{C}_n^{(1)}|\le \rho n\big)
≥P(all ρ2Q2ρn-boxes isolated, ≤ρn vertices)
\ge\,\mathbb{P}\big(\text{all }\frac{2}{\rho}\mathcal{Q}_{\tfrac{\rho n}2}\text{-boxes isolated, $\le\!\rho n$ vertices}\big)
≥(P(Q2ρn-box isolated, ≤ρn vertices))2/ρ
\ge\,\Big(\mathbb{P}\big(\mathcal{Q}_{\tfrac{\rho n}2}\text{-box isolated, $\le\!\rho n$ vertices}\big)\Big)^{2/\rho}
≳exp(−ρ2(2ρn)ζ)
\gtrsim\,\exp\big(-\tfrac{2}\rho(\tfrac{\rho n}2)^\zeta\big)
(FKG)
}
\Bigg\}
n1−ε
n^{1-\varepsilon}
}
\Bigg\}
(δ′logn)1/ζ
(\delta' \log n)^{1/\zeta}
0
0
n
n
E[#small isolated boxes]
\mathbb{E}\big[\# \text{small isolated boxes}\big]
≳nεexp(−(δlogn)1/ζ)ζ)
\gtrsim\, n^\varepsilon \exp\Big(-\big(\delta \log n)^{1/\zeta}\big)^\zeta\Big)
=nε−δ
=\, n^{\varepsilon-\delta}
Lower bounds: ∣Cn(2)∣
P(∣Cn(2)∣≥(δlogn)1/ζ)
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge (\delta \log n)^{1/\zeta}\big)
≥P(∃isolated small box, no long edges)
\ge \, \mathbb{P}\big(\exists \text{isolated small box, no long edges}\big)
⟶∞
\longrightarrow\,\,\,\, \infty
δ small
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),
Upper bound: ∣Cn(2)∣
P(∣Cn(2)∣≥k)≲nexp(−k2−α)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^{2-\alpha}\big)
P(∣Cn(1)∣/n≤ε)≲exp(−n2−α)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \exp\big(-n^{2-\alpha}\big)
P(∃ box without giant)
\mathbb{P}\big(\exists\text{ box without giant}\big)
P(∃ neighboring giants not conn.)
\mathbb{P}\big(\exists\text{ neighboring giants not conn.}\big)
≲(n/k)exp(−k2−α)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
≲(n/k)exp(−k2−α)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
P(∄ backbone)
\mathbb{P}\big(\nexists\text{ backbone}\big)
≲(n/k)exp(−k2−α)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
}
\Bigg\}
k
k
≲(n/k)(1−(2k)−α)(εk)2
\lesssim (n/k)(1-(2k)^{-\alpha})^{(\varepsilon k)^2}
P(size-k component ↔backbone)
\mathbb{P}\big(\text{size-}k\text{ component }\not\leftrightarrow\text{backbone}\big)
≲(n/k)exp(−k2−α)
\lesssim\phantom{(n/k)}\exp\big(-k^{2-\alpha}\big)
P(∃size-k component ↔backbone)
\mathbb{P}\big(\exists \text{size-}k\text{ component }\not\leftrightarrow\text{backbone}\big)
≲(n/k)exp(−k2−α)
\lesssim(n/k)\exp\big(-k^{2-\alpha}\big)
Soft Poisson Boolean model
Long-range percolation
Bond percolation on Zd
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
P(∣Cn(1)∣/n>θ+ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
⋙
\ggg
Lower tail:
- surface tension
- vertex boundary
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
Soft Poisson Boolean model
Edge set E∞
- Long-range parameter α>1,
- Percolation p∈(0,1]
Connection probability
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
Vertex set V∞
- Spatial locations: Poisson point process intensity β>0
- Power-law i.i.d. weights wv≥1:
P(wv≥w)=w−(τ−1),
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
P(u↔v∣V∞)=p(∥xu−xv∥wu1/d+wv1/d∧1)αd
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
Geom. inhom. RG
Long-range percolation
Lower tail*:
If E[#edges of length n1/d]→∞
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
≲exp(−ndd−1∨(2−α))
\lesssim\exp\big(-n^{{\color{darkgray}\frac{d-1}d\vee}(2-\alpha)}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−ndd−1∨(2−α)∨2−(τ−1)/α3−τ)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee\frac{3-\tau}{2-(\tau-1)/\alpha}}\big)
∼exp(−E[∣VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)
Remainder: Lower tail LRP - Upper tail GIRG
*log-corrections at phase transition
Upper tail:
If non-critical, ρ∈(θ,1)
P(∣Cn(1)∣/n>ρ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
∼∗n−I(ρ)
\sim^\ast n^{-I(\rho)}
P(∣Cn(1)∣/n>ρ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
*at continuity points of I(ρ)
⋙
\ggg
⋘
\lll
∼exp(−n)
\sim \exp(-n)
(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)
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
∣Cn(2)∣≳(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
∣Cn(2)∣≲(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}| \lesssim (\log n)^{1/\zeta}
LLN to P(∣C(0)∣=∞)=:ϑ
{\color{blue}\text{LLN }}\text{to }\mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
∣Cn(1)∣ linear
|{\color{blue}\mathcal{C}_n^{(1)}}| \text{ linear}
Lower tail large deviations
{\color{blue}\text{Lower tail large deviations}}
Lower tail of large deviations
{\color{blue}\text{Lower tail of large deviations}}
α
\alpha
τ
\tau
Small
Large
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
∣2nd-largest∣≳(logn)1/ζ
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
∣2nd-largest∣≲(logn)1/ζ
|{\color{red}2^{\mathrm{nd}}\text{-largest}}| \lesssim (\log n)^{1/\zeta}
LLN to P(∣C(0)∣=∞)=:ϑ
{\color{blue}\text{LLN }}\text{to }\mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
∣largest∣ linear
|{\color{blue}\text{largest}}| \text{ linear}
Lower tail large deviations
{\color{blue}\text{Lower tail large deviations}}
Lower tail of large deviations
{\color{blue}\text{Lower tail of large deviations}}
P(k≤∣C(0)∣<∞)≲nexp(−kζ)
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \lesssim \phantom{n}\exp\big(-k^\zeta\big)
∣Cn(2)∣≳(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
P(k≤∣C(0)∣<∞)≳exp(−kζ)
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \gtrsim \exp\big(-k^\zeta\big)
P(∣Cn(2)∣≥k)≲nexp(−kζ)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
∣Cn(1)∣/n⟶P,L1P(∣C(0)∣=∞)=:ϑ
|{\color{blue}\mathcal{C}_n^{(1)}}|\! / n \overset{\mathbb{P}, L^1}\longrightarrow \mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
P(∣Cn(1)∣/n≤ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n≤ϑ−ε)≲exp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta}-\varepsilon\big)\lesssim \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n≤ϑ+ε])≳exp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta} + \varepsilon\big]\big)\gtrsim \exp\big(-n^\zeta\big)
P(k≤∣C(0)∣<∞)≲nexp(−kζ)
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \lesssim \phantom{n}\exp\big(-k^\zeta\big)
∣Cn(2)∣≳(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
P(k≤∣C(0)∣<∞)≳exp(−kζ)
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \gtrsim \exp\big(-k^\zeta\big)
P(∣Cn(2)∣≥k)≲nexp(−kζ)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
∣Cn(1)∣/n⟶P,L1P(∣C(0)∣=∞)=:ϑ
|{\color{blue}\mathcal{C}_n^{(1)}}|\! / n \overset{\mathbb{P}, L^1}\longrightarrow \mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
P(∣Cn(1)∣/n≤ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n≤ϑ−ε)≲exp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta}-\varepsilon\big)\lesssim \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n≤ϑ−ε)≳exp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta} - \varepsilon\big)\gtrsim \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n≤2ε)⟶0
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le 2\varepsilon \big)\longrightarrow 0
Remarks.
- ζ∗<0: subcritical*
-
d≥2:
- (d−1)/d
- partial results: upper bounds
- Product kernel: 2nd term
Power-law degrees: τ>2
d≥2
ζ∗:=max{2−α,(3−τ)/(2−(τ−1)/α)}.
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(3\!-\tau)/(2\!-\!(\tau\!-\!1)/\alpha)
\}.
\end{aligned}
* [Gracar, Lüchtrath, Mönch '22]
Theorem. (J., Komjáthy, Mitsche '23+)
Set
ζ∗:=max{2−α,(τ−2)−(τ−1)/α}.
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(\tau\!-\!2)\!-\!(\tau\!-\!1)/\alpha
\}.
\end{aligned}
If ζ∗>0, then LLN for ∣largest∣, and
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\color{blue}\text{largest}}|\big]}\big)
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∣2nd-largest∣∼(logn)1/ζ∗
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Long-range parameter: α>1
Example 2:
Geom. Inhomog. RG in d=1
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Scale-free Gilbert RG
Random geom. graph
Nearest-neighbor percolation
Large deviations of the giant in spatial random graphs Joost Jorritsma joint with Júlia Komjáthy, Dieter Mitsche Long-range phenomena in percolation, September '24