(Second-)largest component in
spatial random graphs
Dieter Mitsche
joint with Joost Jorritsma, Júlia Komjáthy
Papers: tiny.cc/cluster-size-ksrg
Cluster-size decay in supercrit. nearest-neighbor percolation
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)=
\exp\big(-\Theta(\sqrt{k})\big)
[Grimmett & Marstrand '90, Kesten & Zhang '90]
Cluster-size decay in supercrit. nearest-neighbor percolation
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)=
\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
\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|\ge k, 0\notin\mathcal C_n^{(1)}\big)=
\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 \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
Edge set \(\mathcal{E}_\infty\)
- Long-range parameter \(\alpha>1\),
- Edge-density \(\beta>0\),
- Percolation \(p\in(0,1]\)
Connection probability
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{\beta}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{\beta}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{\beta}{\|x_u-x_v\|^d}\phantom{\wedge 1}\bigg)^\alpha$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p\bigg(}\frac{1}{\|x_u-x_v\|^d}\phantom{\bigg)^\alpha\wedge 1}$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{1}{\|x_u-x_v\|^d}\phantom{\wedge 1}\bigg)^\alpha$$
Kernel-based spatial random graphs
Vertex set \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
- Power-law i.i.d. weights \(w_v\ge 1\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),
Kernel-based spatial random graphs
Edge set \(\mathcal{E}_\infty\)
- Long-range parameter \(\alpha>1\),
- Edge-density \(\beta>0\),
- Percolation \(p\in(0,1]\)
Connection probability
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta\cdot \phantom{(w_u\cdot w_v)}}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
Vertex set \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
- Power-law i.i.d. weights \(w_v\ge 1\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta\cdot (w_u\cdot w_v)}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
Kernel-based spatial random graphs
Edge set \(\mathcal{E}_\infty\)
- Long-range parameter \(\alpha>1\),
- Edge-density \(\beta>0\),
- Percolation \(p\in(0,1]\)
Connection probability
Vertex set \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
- Power-law i.i.d. weights \(w_v\ge 1\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta\cdot (w_u\cdot w_v)}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$
Components in supercritical graphs
-
Largest component \({\color{blue}\mathcal{C}_n^{(1)}}\):
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations
Questions
Components in supercritical graphs
-
Largest component \({\color{blue}\mathcal{C}_n^{(1)}}\):
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations
Questions
Previous results
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\lesssim(\log n)^{??}
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}d/(d-1)}
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
\sim \exp\!\big(\!\!-\!k^{\color{darkred}(d-1)/d}\big)
\sim \exp\big(\!-\!n^{\color{darkred} (d-1)/d}\big)
\lesssim \exp\big(\!-\!n^{\max(2-\alpha-o(1), (d-1)/d)}\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\big)
\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]
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}2/(3-\tau)}
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\big)
\lesssim \exp\big(\!-\!n^{\color{darkred}(3-\tau)/2}\big)
Conjecture: \(\exists \zeta\in \big[\tfrac{d-1}{d},1\big)\):
\sim \exp\big(\!-\!n^{\color{darkred}\zeta}\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\color{blue}\mathrm{largest}}|\big]}\big)
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta}\big)
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta}
Theorem. Long-range percolation (J., Komjáthy, Mitsche '24+)
Set \(\zeta_\ast := \max\{\zeta_\mathrm{long}, (d-1)/d\}\), with \(\zeta_\mathrm{long}=2-\alpha\).
If \(d=1\) and \(\zeta_\ast>0\).
If \(d\ge2\), and
\(0<\zeta_\mathrm{long}\le(d-1)/d\).
\(\zeta_\mathrm{long}>(d-1)/d\).
\(\zeta_\mathrm{long}\le(d-1)/d\), and \(p\) or \(\beta\) large.
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
\mathbb{P}\big(|{\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\mathrm{largest}}|\big]}\big)
\mathbb{P}\big(k\!\le\! |{\mathcal{C}(0)}|\!<\!\infty\big)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
|{2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Conjecture.
Theorem. Geom. Inhom. RG (J., Komjáthy, Mitsche '24+)
Set \(\zeta_\ast := \max\{\zeta_\mathrm{long}, (d-1)/d\}\), with \(\zeta_\mathrm{long}=\max\Big\{2-\alpha, \frac{3-\tau}{2-(\tau-1)/\alpha}\Big\}\).
If \(d=1\) and \(\zeta_\ast>0\).
If \(d\ge2\), and
\(0<\zeta_\mathrm{long}\le(d-1)/d\).
\(\zeta_\mathrm{long}>(d-1)/d\).
\(\zeta_\mathrm{long}\le(d-1)/d\), and \(p\) or \(\beta\) large.
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
\mathbb{P}\big(|{\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\mathrm{largest}}|\big]}\big)
\mathbb{P}\big(k\!\le\! |{\mathcal{C}(0)}|\!<\!\infty\big)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
|{2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Conjecture.
{\color{green}\text{Cluster-size decay}}
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
{\color{green}\text{Cluster-size decay}}
|{\color{red}2^{\mathrm{nd}}\text{-largest}}| \lesssim (\log n)^{1/\zeta}
{\color{blue}\text{LLN }}\text{to }\mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
|{\color{blue}\text{largest}}| \text{ linear}
{\color{blue}\text{Lower tail large deviations}}
{\color{blue}\text{Lower tail of large deviations}}
Lower bounds: cluster-size decay
Aim: Find minimal \(\zeta\) s.t.
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);
\mathbb{P}\big(k\le |\mathcal{C}(0)|<\infty\big)
\sim\mathbb{P}\big(\mathcal{Q}_k\text{ isolated}\big)
\sim \exp\big(-k^\zeta\big)
|\mathcal{C}(0)|\ge k
\Bigg\}
\Theta(k)
\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]
Aim: Find \(\gamma\) s.t.
\sim k^\zeta
Lower bounds: cluster-size decay
Aim: Find minimal \(\zeta\) s.t.
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);
|\mathcal{C}(0)|\ge k
\Bigg\}
\Theta(k)
k^\zeta\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\searrow \Lambda_k^c\}|\big]
\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]
Aim: Find \(\gamma\) s.t.
\sim k^\zeta
Lower bounds: large deviations
0
n
\Bigg\}
\rho n/2
\mathbb{P}\big(|\mathcal{C}_n^{(1)}|\le \rho n\big)
\ge\,\mathbb{P}\big(\text{all }\frac{2}{\rho}\mathcal{Q}_{\tfrac{\rho n}2}\text{-boxes isolated, $\le\!\rho n$ vertices}\big)
\ge\,\Big(\mathbb{P}\big(\mathcal{Q}_{\tfrac{\rho n}2}\text{-box isolated, $\le\!\rho n$ vertices}\big)\Big)^{2/\rho}
\gtrsim\,\exp\big(-\tfrac{2}\rho(\tfrac{\rho n}2)^\zeta\big)
(FKG)
\Bigg\}
n^{1-\varepsilon}
\Bigg\}
(\delta' \log n)^{1/\zeta}
0
n
\mathbb{E}\big[\# \text{small isolated boxes}\big]
\gtrsim\, n^\varepsilon \exp\Big(-\big(\delta \log n)^{1/\zeta}\big)^\zeta\Big)
=\, n^{\varepsilon-\delta}
Lower bounds: \(|\mathcal{C}_n^{(2)}|\)
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge (\delta \log n)^{1/\zeta}\big)
\ge \, \mathbb{P}\big(\exists \text{isolated small box, no long edges}\big)
\longrightarrow\,\,\,\, \infty
\(\delta\) small
{\color{green}\text{Cluster-size decay}}
Upper bounds
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
\text{Law of large numbers}
{\text{\color{blue}Lower tail large deviations}}
Challenge: Delocalized components
\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)
# possibilities for \(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\)
Upper bound: \(|\mathcal{C}_n^{(2)}|\)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^{2-\alpha}\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \exp\big(-n^{2-\alpha}\big)
\mathbb{P}\big(\exists\text{ box without giant}\big)
\mathbb{P}\big(\exists\text{ neighboring giants not conn.}\big)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
\mathbb{P}\big(\nexists\text{ backbone}\big)
\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)
\Bigg\}
k
\lesssim (n/k)(1-(2k)^{-\alpha})^{(\varepsilon k)^2}
\mathbb{P}\big(\text{size-}k\text{ component }\not\leftrightarrow\text{backbone}\big)
\lesssim\phantom{(n/k)}\exp\big(-k^{2-\alpha}\big)
\mathbb{P}\big(\exists \text{size-}k\text{ component }\not\leftrightarrow\text{backbone}\big)
\lesssim(n/k)\exp\big(-k^{2-\alpha}\big)
\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
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta:=\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|=\infty\big)
{\text{\color{blue}Lower tail large deviations}}
Truncation
Local convergence: giant is almost local
Lower tail of large deviations
|{\color{blue}\mathcal{C}_n^{(1)}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|\le \vartheta-\varepsilon\big)\lesssim \exp\big(-n^{2-\alpha}\big)
\Bigg\}
n^{\alpha-1}
\(m_n\sim n^{2-\alpha}\) boxes
\mathbb{P}\big(|\mathcal{C}_{n^{\alpha-1}}^{(1)}|/n^{\alpha-1} \le \vartheta - \varepsilon\big)\le \varepsilon/2
\mathbb{P}\big(\text{more than }\varepsilon m_n\text{ non-giants}\big)\lesssim\exp\big(-n^{2-\alpha}\big)
\mathbb{P}\big(\text{giants connect}\big)\sim
n^{\alpha-1}n^{\alpha-1}n^{-\alpha}
\sim n^{-(2-\alpha)}
\ge \lambda/m_n
\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
|{\color{blue}\mathcal{C}_n^{(1)}}|/n \overset{\mathbb{P}}\longrightarrow \vartheta
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|\le \vartheta-\varepsilon\big)\lesssim \exp\big(-n^{2-\alpha}\big)
\(m_n\sim n^{2-\alpha}\) boxes
\mathbb{P}\big(\text{giants connect}\big)\ge \lambda/m_n
\mathbb{P}\big(\text{ giant in }\text{ER}(m, \lambda/m)\text{ has size at least }\rho(\lambda)-\varepsilon\big)
\sim \exp\big(-m\big)
\mathbb{P}\big(\text{more than }\varepsilon m_n\text{ non-giants}\big)\lesssim\exp\big(-n^{2-\alpha}\big)
\Bigg\}
n/m_n
{\color{green}\text{Cluster-size decay}}
Upper bounds
\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
\text{Law of large numbers}
{\text{\color{blue}Lower tail large deviations}}
What about other value \(\zeta_\ast\)?
\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 \({\color{blue}\mathcal{C}_n^{(1)}}\):
- 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
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\rho\big)\sim \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)\sim
n^{-(\tau-2)\lceil h(\rho)\rceil}
\mathbb{E}\Big[(1-p)^{h(\rho)|{\color{green}\mathcal{C}(0)}|}\Big]= 1-\rho
Lower tail:
Upper tail:
# hubs (\(w_v\sim n\)) required: increase density from \(\vartheta\) to \(\vartheta+\varepsilon\)
(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 \(d\ge 2\)
- Extension from PPP to grid
- Central limit theorem
- \(\zeta\in\big[(d-1)/d, 1\big)\)
Thank you!
{\color{green}\text{Cluster-size decay}}
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
{\color{green}\text{Cluster-size decay}}
|{\color{red}\mathcal{C}_n^{(2)}}| \lesssim (\log n)^{1/\zeta}
{\color{blue}\text{LLN }}\text{to }\mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
|{\color{blue}\mathcal{C}_n^{(1)}}| \text{ linear}
{\color{blue}\text{Lower tail large deviations}}
{\color{blue}\text{Lower tail of large deviations}}
\alpha
\tau
Small
Large
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \lesssim \phantom{n}\exp\big(-k^\zeta\big)
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \gtrsim \exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
|{\color{blue}\mathcal{C}_n^{(1)}}|\! / n \overset{\mathbb{P}, L^1}\longrightarrow \mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta}-\varepsilon\big)\lesssim \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta} + \varepsilon\big]\big)\gtrsim \exp\big(-n^\zeta\big)
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \lesssim \phantom{n}\exp\big(-k^\zeta\big)
|{\color{red}\mathcal{C}_n^{(2)}}|\gtrsim (\log n)^{1/\zeta}
Lower bounds
Upper bounds
\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big) \gtrsim \exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{red}\mathcal{C}_n^{(2)}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
|{\color{blue}\mathcal{C}_n^{(1)}}|\! / n \overset{\mathbb{P}, L^1}\longrightarrow \mathbb{P}(|{\color{green}\mathcal{C}(0)}|\!=\!\infty)=:{\color{green}\vartheta}
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta}-\varepsilon\big)\lesssim \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le {\color{green}\vartheta} - \varepsilon\big)\gtrsim \exp\big(-n^\zeta\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le 2\varepsilon \big)\longrightarrow 0
Remarks.
- \(\zeta_\ast<0\): subcritical*
-
\(d\ge 2\):
- \((d-1)/d\)
- partial results: upper bounds
- Product kernel: 2nd term
Power-law degrees: \(\tau>2\)
\(d\ge 2\)
\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
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(\tau\!-\!2)\!-\!(\tau\!-\!1)/\alpha
\}.
\end{aligned}
If \(\zeta_\ast>0\), then LLN for \(|{\color{blue}\text{largest}}|\), and
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! (1-\varepsilon ){\mathbb{E}\big[|{\color{blue}\text{largest}}|\big]}\big)
\mathbb{P}\big(k\!\le\! |{\color{green}\mathcal{C}(0)}|\!<\!\infty\big)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
\sim \exp\big(\!-\!n^{\color{darkred}\zeta_\ast}\big)
|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\sim(\log n)^{\color{darkred}1/\zeta_\ast}
Long-range parameter: \(\alpha>1\)
Example 2:
Geom. Inhomog. RG in \(d=1\)
(Second-)largest component in spatial inhomogeneous random graphs - Dieter
By joostjor
