(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
P(k≤∣C(0)∣<∞)=
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)=
exp(−Θ(k))
\exp\big(-\Theta(\sqrt{k})\big)
[Grimmett & Marstrand '90, Kesten & Zhang '90]
Cluster-size decay in supercrit. nearest-neighbor percolation
P(k≤∣C(0)∣<∞)=
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)=
exp(−Θ(kζ)),ζ=dd−1
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big),\qquad\zeta=\frac{d-1}d
[Grimmett & Marstrand '90, Kesten & Zhang '90]
Cluster-size decay in Erdős–Rényi random graph
P(∣C(0)∣≥k,0∈/Cn(1))=
\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|\ge k, 0\notin\mathcal C_n^{(1)}\big)=
exp(−Θ(kζ)),ζ=1
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big),\qquad\zeta=1
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
∣2nd-largest∣≲(logn)??
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\lesssim(\log n)^{??}
∣2nd-largest∣∼(logn)d/(d−1)
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}d/(d-1)}
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
∼exp(−k(d−1)/d)
\sim \exp\!\big(\!\!-\!k^{\color{darkred}(d-1)/d}\big)
∼exp(−n(d−1)/d)
\sim \exp\big(\!-\!n^{\color{darkred} (d-1)/d}\big)
≲exp(−nmax(2−α−o(1),(d−1)/d))
\lesssim \exp\big(\!-\!n^{\max(2-\alpha-o(1), (d-1)/d)}\big)
P(∣largest∣≤εn)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\big)
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon)\mathbb{E}[|{\color{blue}\mathrm{largest}}|]\big)
[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]
∣2nd-largest∣∼(logn)2/(3−τ)
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}2/(3-\tau)}
P(∣largest∣≤εn)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\big)
≲exp(−n(3−τ)/2)
\lesssim \exp\big(\!-\!n^{\color{darkred}(3-\tau)/2}\big)
Conjecture: ∃ζ∈[dd−1,1):
∼exp(−nζ)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta}\big)
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\color{blue}\mathrm{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}\big)
∣2nd-largest∣∼(logn)1/ζ
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta}
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.
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\mathrm{largest}}|\big]}\big)
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\!\le\! |{\mathcal{C}(0)}|\!<\!\infty\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∣2nd-largest∣∼(logn)1/ζ∗
|{2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
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.
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
P(∣largest∣≤(1−ε)E[∣largest∣])
\mathbb{P}\big(|{\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\mathrm{largest}}|\big]}\big)
P(k≤∣C(0)∣<∞)
\mathbb{P}\big(k\!\le\! |{\mathcal{C}(0)}|\!<\!\infty\big)
∼exp(−nζ∗)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
∣2nd-largest∣∼(logn)1/ζ∗
|{2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Conjecture.
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}}
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
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
Upper bounds
P(∣2nd-largest∣≥k)≲nexp(−kζ)
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
P(∣largest∣/n≤ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
Law of large numbers
\text{Law of large numbers}
Lower tail large deviations
{\text{\color{blue}Lower tail large deviations}}
Challenge: Delocalized components
(kn)∼nk≫nexp(kζ)
\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)
# possibilities for ∣2nd-largest∣≥k
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)
P(∣C(0)∣≥k,0∈/largest)≲nexp(−kζ)
\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|\ge k, 0\notin{\color{blue}\text{largest}}\big)\lesssim \phantom{n}\exp\big(-k^\zeta\big)
Upper bounds
P(∣2nd-largest∣≥k)≲nexp(−kζ)
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
P(∣largest∣/n≤ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
∣largest∣/n⟶Pϑ:=P(∣C(0)∣=∞)
|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta:=\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|=\infty\big)
Lower tail large deviations
{\text{\color{blue}Lower tail large deviations}}
Truncation
Local convergence: giant is almost local
Lower tail of large deviations
∣Cn(1)∣/n⟶Pϑ
|{\color{blue}\mathcal{C}_n^{(1)}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta
P(∣Cn(1)∣≤ϑ−ε)≲exp(−n2−α)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|\le \vartheta-\varepsilon\big)\lesssim \exp\big(-n^{2-\alpha}\big)
}
\Bigg\}
nα−1
n^{\alpha-1}
mn∼n2−α boxes
P(∣Cnα−1(1)∣/nα−1≤ϑ−ε)≤ε/2
\mathbb{P}\big(|\mathcal{C}_{n^{\alpha-1}}^{(1)}|/n^{\alpha-1} \le \vartheta - \varepsilon\big)\le \varepsilon/2
P(more than εmn non-giants)≲exp(−n2−α)
\mathbb{P}\big(\text{more than }\varepsilon m_n\text{ non-giants}\big)\lesssim\exp\big(-n^{2-\alpha}\big)
P(giants connect)∼
\mathbb{P}\big(\text{giants connect}\big)\sim
nα−1nα−1n−α
n^{\alpha-1}n^{\alpha-1}n^{-\alpha}
∼n−(2−α)
\sim n^{-(2-\alpha)}
≥λ/mn
\ge \lambda/m_n
E[{v:v has edge of length Ω(n1/d)}]=Θ(n2−α)
\mathbb{E}\Big[\big|\big\{v: v \text{ has edge of length }\Omega(n^{1/d})\big\}\big|\Big] = \Theta(n^{2-\alpha})
Lower tail of large deviations
∣Cn(1)∣/n⟶Pϑ
|{\color{blue}\mathcal{C}_n^{(1)}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta
P(∣Cn(1)∣≤ϑ−ε)≲exp(−n2−α)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|\le \vartheta-\varepsilon\big)\lesssim \exp\big(-n^{2-\alpha}\big)
mn∼n2−α boxes
P(giants connect)≥λ/mn
\mathbb{P}\big(\text{giants connect}\big)\ge \lambda/m_n
P( giant in ER(m,λ/m) has size at least ρ(λ)−ε)
\mathbb{P}\big(\text{ giant in }\text{ER}(m, \lambda/m)\text{ has size at least }\rho(\lambda)-\varepsilon\big)
∼exp(−m)
\sim \exp\big(-m\big)
P(more than εmn non-giants)≲exp(−n2−α)
\mathbb{P}\big(\text{more than }\varepsilon m_n\text{ non-giants}\big)\lesssim\exp\big(-n^{2-\alpha}\big)
}
\Bigg\}
n/mn
n/m_n
Cluster-size decay
{\color{green}\text{Cluster-size decay}}
Upper bounds
P(∣2nd-largest∣≥k)≲nexp(−kζ)
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
P(∣largest∣/n≤ε)≲nexp(−nζ)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
Law of large numbers
\text{Law of large numbers}
Lower tail large deviations
{\text{\color{blue}Lower tail large deviations}}
What about other value ζ∗?
ζ∗:=max{2−α,(3−τ)/(2−(τ−1)/α)}
\zeta_\ast := \max\{2\!-\!\alpha, (3\!-\!\tau)\big/(2\!-\!(\tau\!-\!1)/\alpha)
\}
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
P(∣Cn(1)∣/n<ρ)∼exp(−nζ)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\rho\big)\sim \exp\big(-n^\zeta\big)
P(∣Cn(1)∣/n>ρ)∼
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)\sim
n−(τ−2)⌈h(ρ)⌉
n^{-(\tau-2)\lceil h(\rho)\rceil}
E[(1−p)h(ρ)∣C(0)∣]=1−ρ
\mathbb{E}\Big[(1-p)^{h(\rho)|{\color{green}\mathcal{C}(0)}|}\Big]= 1-\rho
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!
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
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
(Second-)largest component in spatial random graphs Joost Jorritsma joint with Júlia Komjáthy, Dieter Mitsche Papers: tiny.cc/cluster-size-ksrg
Copy of (Second-)largest component in spatial inhomogeneous random graphs
By joostjor
Copy of (Second-)largest component in spatial inhomogeneous random graphs
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