Large deviations of the giant in spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche
LA Probability Forum, April '25
Supercritical Erdős–Rényi random graph G(n,λ/n)
P(∣C(0)∣>k,C(0)=Cn(1))∼
\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|>k, {\color{green}\mathcal{C}(0)}\neq {\color{blue}\mathcal{C}_n^{(1)}}\big)\sim
exp(−Θ(kζ))
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big)
[Erdős, Rényi '59; O'Connell '98; Andreis, König, Langhammer, Patterson'23]
∣Cn(2)∣∼P(logn)1/ζ
|{\color{red}\mathcal{C}_n^{(2)}}|\sim_\mathbb{P}(\log n)^{1/\zeta}
Exponential growth of neighbourhood
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(λ):=P(Bienaymeˊ(Poiλ) survives)>0
\theta(\lambda):=\mathbb{P}(\text{Bienaymé(Poi}_\lambda)\text{ survives})>0
ζ=1
\zeta=1
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)
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)
[Alexander, Chayes, Chayes '90; Cerf '00; 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 in volume-n box
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}[|\searrow\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)∨((τ−1)/α−(τ−2)))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee((\tau-1)/\alpha-(\tau-2))}\big)
∼exp(−E[∣↘VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\searrow\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<\theta- \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
Goal 1: Law of large numbers
+ local convergence (Giant is almost local)
[vdHofstad, vdHoorn, Maitra '21, vdHofstad '21]
Goal 2: Prove the theorem
k→∞limn→∞limP0(∣Cn(0)∣>k,0∈/Cn(1))=0
∣Cn(2)∣=Θ(polylog(n))
P(∣largest∣/n<ε)≲nexp(−nζ−o(1))
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\varepsilon \big)\lesssim \phantom{n} \exp\big(-n^{\zeta-o(1)}\big)
P(∣largest∣/n<ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\varepsilon \big)\lesssim \phantom{n} \exp\big(-n^{\zeta}\big)
Edge set E∞
- Long-range parameter α>1,
- Percolation p∈(0,1]
Connection probability
Vertex set V∞
- Locations: Poisson point process intensity β>0
- Constant weights
P(u↔v∣V∞)≈∥xu−xv∥αdp
P(∣largest∣/n<θ−ε)≲exp(−n2−α)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\theta- \varepsilon \big)\lesssim \exp\big(-n^{2-\alpha}\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))
\approx\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
≈exp(−E[∣↘VB(n)∣])
\approx\exp\big(-\mathbb{E}[|\searrow\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)∨((τ−1)/α−(τ−2)))
\approx\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee((\tau-1)/\alpha-(\tau-2))}\big)
≈exp(−E[∣↘VB(n)∣])
\approx\exp\big(-\mathbb{E}[|\searrow\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)
\approx^\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)
\approx \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
Power-law weights
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}[|\searrow\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)∨((τ−1)/α−(τ−2))
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee(
(\tau-1)/\alpha-(\tau-2)}\big)
∼exp(−E[∣↘VB(n)∣])
\sim\exp\big(-\mathbb{E}[|\searrow\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)
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)
[Alexander, Chayes, Chayes '90; Cerf '00; 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)
Lower tail large deviations of the giant*:
P(∣Cn(1)∣/n<θ−ε)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
∼exp(−nζ)
\sim\exp\big(-n^{\zeta}\big)
One decay exponent governing cluster sizes
*log(log)-corrections at phase transition
∣Cn(2)∣
|{\color{red}\mathcal{C}_n^{(2)}}|
∼(logn)1/ζ
\sim (\log n)^{1/\zeta}
Small components*:
P(k<∣C(0)∣<∞)
\mathbb{P}\big(k<|{\color{green}\mathcal{C}(0)}|<\infty\big)
∼exp(−nζ)
\sim \exp\big(-n^{\zeta}\big)
ζ:=limn→∞lognlogE[∣“downward”-vertex-boundary(n)∣]
\zeta:=\lim_{n\to\infty}\frac{\log\mathbb{E}[|\text{``downward''-vertex-boundary}(n)|]}{\log n}
If E[#edges of length n1/d]→∞
If ζ>(d−1)/d
(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.
- Extension from PPP to grid
- Partial results d≥2
- Rate functions (lower tail)
- Central limit theorem
- ζ∈[(d−1)/d,1)
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;
arXiv: 2303.00712, Annals of Probability (to appear). -
Large deviations of the giant in supercritical KSRGs;
With Júlia Komjáthy, Dieter Mitsche;
Preprint arXiv: 2404.02984.
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
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)
(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}}
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
Large deviations of the giant in spatial random graphs Joost Jorritsma joint with Júlia Komjáthy, Dieter Mitsche LA Probability Forum, April '25
Copy of Largest component in spatial random graphs
By joostjor
Copy of Largest component in spatial random graphs
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