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 \(\mathcal{V}_\infty\)
-
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 \(\mathcal{E}_\infty\)
- Symmetric kernel \(\kappa(w_u, w_v)\),
- Long-range parameter \(\alpha\in(1,\infty]\),
- Edge-density \(\beta>0\),
Connection probability
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$$
Vertex set \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
- Power-law i.i.d. weights \(w_v\ge 1\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{\kappa(w_u, w_v)}{\phantom{\|x_u-x_v\|^d}}\phantom{\bigg)^\alpha\wedge 1}$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\phantom{\bigg)^\alpha\wedge 1}$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\phantom{\beta}\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\phantom{\wedge 1}$$
Kernel-based spatial random graphs
Edge set \(\mathcal{E}_\infty\)
- Symmetric kernel \(\kappa(w_1, w_2)\),
- Edge-density \(\beta>0\),
- Long-range parameter \(\alpha\in(1,\infty]\),
Connection probability
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$$
Vertex set \(\mathcal{V}_\infty\)
-
Spatial locations, either
- Lattice \(\mathbb{Z}^d\)
- Poisson point process (unit intensity)
- Power-law i.i.d. weights \(w_v\ge 1\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),
\kappa_\mathrm{prod}=w_uw_v
\alpha\in(1,\infty], d\ge 1
Geom. Inhom. RG
\kappa_\mathrm{prod}=w_uw_v
\alpha=\infty, d= 1
Hyperbolic RG
Geom. RG
\kappa_\mathrm{triv}\equiv 1
\alpha=\infty, d \ge 2
Long-range perc.
\kappa_\mathrm{triv}\equiv 1
\alpha\in(1,\infty), d\ge 1
\kappa_{\max}=w_u\vee w_v
Scale-free Gilbert RG
\kappa_{\mathrm{pa}}=(w_u\vee w_v)
\cdot(w_u\wedge w_v)^{\tau-2}
Age-dependent RCM
Theorem. When \(\tau<2\) or \(\alpha<1\):
- 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 \(\tau>2\) and \(\alpha>1\):
$$\mathbb{P}(\mathrm{deg}(0)\ge k)\sim k^{-(\tau-1)}.$$
\(\tau\) small: many hubs
\(\alpha\) small: many long edges
The interpolating kernel
Connection probability
$${\color{grey}\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta}\frac{\kappa(w_u, w_v)}{\color{grey}\|x_u-x_v\|^d}{\color{grey}\bigg)^\alpha\wedge 1}$$
A parameterized kernel: \(\sigma\ge 0\)
$$\kappa_{\sigma}(w_u, w_v):=\max\{w_u, w_v\}\min\{w_u, w_v\}^\sigma$$
- \(\tau: \mathbb{P}(w_v\ge w)=w^{-(\tau-1)}.\)
- \(\sigma\): assortativity
- \(\sigma\): interpolation
\frac{\sigma}{\tau-1}
\frac1{\tau-1}
1
1
\sigma=\tau-1
\tau=2
\sigma=1
\sigma=1
\sigma=\tau-2
\sigma=\tau-2
\sigma=0
\sigma=0
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 \({\color{blue}\mathcal{C}_n^{(1)}}\):
- 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}2/(3-\tau)}
|{\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)
\lesssim \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)
\lesssim \exp\big(\!-\!n^{\color{darkred}(3-\tau)/2}\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|\!\le\! \varepsilon n\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]
Lower tail large deviations of the giant*:
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\sim\exp\big(-n^{\zeta}\big)
One decay exponent governing cluster sizes
*log(log)-corrections at phase transition
|{\color{red}\mathcal{C}_n^{(2)}}|
\sim (\log n)^{1/\zeta}
Small components*:
\mathbb{P}\big(k<|{\color{green}\mathcal{C}(0)}|<\infty\big)
\sim \exp\big(-n^{\zeta}\big)
\zeta:=\lim_{n\to\infty}\frac{\log\mathbb{E}[|\text{``downward''-vertex-boundary}(n)|]}{\log n}
If \(\mathbb{E}[\# \text{edges of length } \Omega(n^{1/d})]\to\infty\)
If \(\zeta>(d-1)/d\)
{\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}}
|{\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}}
{\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)
Challenge: Delocalized components
\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)
# possibilities for \(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\)
\text{{\color{blue}largest} linear w.h.p.}
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)
What about other values \(\zeta_\ast\)?
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(3\!-\!\tau)\big/(2\!-\!(\tau\!-\!1)/\alpha)
\}.
\end{aligned}
\begin{aligned}
\zeta_\ast := \max\{&2\!-\!\alpha,\\
&(\tau\!-\!1)/\alpha - (\tau-2))
\}.
\end{aligned}
\(\kappa_\mathrm{prod}=w_u w_v\)
\(\kappa_\mathrm{max}=w_u\vee w_v\)
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 \(d\ge 2\)
- Extension from PPP to grid
- Central limit theorem
- \(\zeta\in\big[(d-1)/d, 1\big)\)
Thank you!
Supercritical Erdős–Rényi random graph \(G(n, \lambda/n)\)
\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|>k, {\color{green}\mathcal{C}(0)}\neq {\color{blue}\mathcal{C}_n^{(1)}}\big)\sim
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big)
[Erdős, Rényi '59; O'Connell '98; Andreis, König, Langhammer, Patterson'23]
|{\color{red}\mathcal{C}_n^{(2)}}|\sim_\mathbb{P}(\log n)^{1/\zeta}
Exponential growth of neighbourhood
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)\sim
\exp\big(-\Theta(n^{\mathbf{\zeta}})\big)
\theta(\lambda):=\mathbb{P}(\text{Bienaymé(Poi}_\lambda)\text{ survives})>0
\zeta=1
The largest component
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)\sim
\exp\big(-\Theta(n)\big)
Supercritical bond percolation on \(\mathbb{Z}^d\)
\mathbb{P}\big(k< |{\color{green}\mathcal{C}(0)}|<\infty\big)\sim
\exp\big(-\Theta(k^{\mathbf{\zeta}})\big)
[Alexander, Chayes, Chayes '90; Cerf '00; Gandolfi '88; Grimmett & Marstrand '90, Kesten & Zhang '90, Pisztora '96]
|{\color{red}\mathcal{C}_n^{(2)}}|\sim_\mathbb{P}(\log n)^{1/\zeta}
Surface-tension driven behavior
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)\sim
\exp\big(-\Theta(n^{\mathbf{\zeta}})\big)
\theta(p):=\mathbb{P}(0\leftrightarrow \infty)>0
\zeta=\frac{d-1}d
The largest component
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)\sim
\exp\big(-\Theta(n)\big)
Remarks.
- \(\zeta_\ast<0\): subcritical*
-
\(d\ge 2\):
- \((d-1)/d\)
- partial results: upper bounds
- Product kernel: 2nd term
No known results
- \(|{\color{blue}\text{largest}}|/n\)
- \({\color{red}2^{\mathrm{nd}}\text{-largest}}\)
Example 1:
Scale-free Gilbert RG in \(d=1\)
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: \(\alpha>1\)
\(\kappa_\mathrm{max}=w_u\vee w_v\)
\(\kappa_\mathrm{prod}=w_uw_v\)
Example 2:
Geom. Inhomog. RG in \(d=1\)
{\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 \(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\)
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}
{\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}}
{\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 (\(d=1\))
(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
- \(\zeta\in\big[(d-1)/d, 1\big)\)
{\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
{\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}}
\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 \(d=1\)
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Scale-free Gilbert RG
Random geom. graph
Nearest-neighbor percolation
Copy of Largest component in spatial random graphs
By joostjor
