Large deviations of the giant in spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche
RandNET Workshop, September '25
Puerto Sonoro
Cumbia, Ska, Funk, Reggea
Friday, 20:30 (Club 1019), €10.55
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 in volume-\(n\) box
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)\sim
\exp\big(-\Theta(n)\big)
Soft Poisson Boolean model
Edge set \(\mathcal{E}_\infty\)
- \(\{u\leftrightarrow v \}\Longleftrightarrow \{w_u^{1/d}+w_v^{1/d}\ge \|x_u-x_v\|\}\)
- Long-range parameter \(\alpha>1\),
- Percolation \(p\in(0,1]\)
Connection probability
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p\bigg(}\frac{ w_u^{1/d}+ w_v^{1/d}}{\|x_u-x_v\|}\wedge 1\phantom{\bigg)^{\alpha d}}$$
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{ w_u^{1/d}+ w_v^{1/d}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$
Vertex set \(\mathcal{V}_\infty\)
- Locations: Poisson point process intensity \(\beta>0\)
- Power-law i.i.d. weights \(\tau>2\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\), \(w\ge 1\)
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{w_u^{1/d}+ w_v^{1/d}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$
$$(\mathrm{deg}(v) \mid w_v=w) \, = \, \mathrm{Poi}(c_{\alpha, d, \tau}p\beta \cdot w)$$
Soft Poisson Boolean model
Supercriticality assumption
- \(\theta=\theta_{d, \tau, \alpha, p}(\beta):=\mathbb{P}_{d, \tau, \alpha, p, \beta}^0(0\leftrightarrow \infty)\)
- \(\beta>\beta_c(\alpha, \tau, d, p):=\inf\{\beta: \theta(\beta)>0\}\)
Edge set \(\mathcal{E}_\infty\)
- \(\{u\leftrightarrow v \}\Longleftrightarrow \{w_u^{1/d}+w_v^{1/d}\ge \|x_u-x_v\|\}\)
- Long-range parameter \(\alpha>1\),
- Percolation \(p\in(0,1]\)
Vertex set \(\mathcal{V}_\infty\)
- Locations: Poisson point process intensity \(\beta>0\)
- Power-law i.i.d. weights \(\tau>2\):
\(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\), \(w\ge 1\)
Soft Poisson Boolean model
Questions:
Fixed parameters:
- Point process on \(\mathbb{R}^d\) intensity \(\beta>0\)
- Power-law exponent \(\tau>2\)
- Long-range parameter \(\alpha>1\)
- Percolation \(p\in(0,1]\)
[Gracar, Lüchtrath, Mönch '21]
- When is \(\beta_c<\infty\)?
- \(d\ge 2\),
- Size of the largest connected component \(|\mathcal{C}_n^{(1)}|\)
- Law of large numbers (\(|\mathcal{C}_n^{(1)}|/n\to \theta\))?
- surface-tension behav. in \(\mathbb{P}(|\mathcal{C}_n^{(1)}|/n<\theta-\varepsilon)\)?
- exponential decay of \(\mathbb{P}(|\mathcal{C}_n^{(1)}|/n>\theta+\varepsilon)\)?
or \(d=1\), and \(\alpha\in(1, 2\vee\alpha_\tau)\)
Power-law weights
Constant weights
Lower tail*:
If \(\mathbb{E}[\# \text{edges of length } n^{1/d}]\to\infty\)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
\sim\exp\big(-\mathbb{E}[|\searrow\mathrm{VB}(n)|]\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee((\tau-1)/\alpha-(\tau-2))}\big)
\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
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
\sim^\ast n^{-I(\theta+\varepsilon)}
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
*at continuity points of \(I(\theta+\varepsilon)\)
\ggg
\lll
\sim \exp(-n)
Biskup ['04]
\mathbb{P}(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \varepsilon)
\lesssim\exp\big(-n^{2-\alpha-o(1)}\big)
Lower bound for the lower tail
Aim: Find minimal \(\zeta\) s.t.
\mathbb{P}\big(\mathcal{Q}_k\text{ isolated}\big)\gtrsim\exp\big(-k^\zeta\big);
\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
\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\searrow \Lambda_k^c\}|\big]
\mathbb{P}\big(\mathcal{Q}_k\text{ isolated}\big)\gtrsim\exp\big(-\mathbb{E}[|\phantom{\square\square}|]\big)
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)
Power-law weights
Constant weights
Lower tail*:
If \(\mathbb{E}[\# \text{edges of length } n^{1/d}]\to\infty\)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\approx\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
\approx\exp\big(-\mathbb{E}[|\searrow\mathrm{VB}(n)|]\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\approx\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee((\tau-1)/\alpha-(\tau-2))}\big)
\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
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
\approx^\ast n^{-I(\theta+\varepsilon)}
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)
*at continuity points of \(I(\theta+\varepsilon)\)
\ggg
\lll
\approx \exp(-n)
Upper tail: polynomial decay when \(\tau<\infty\)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)=
Related work:
- Sum of iid Par(\(\tau)\): principle of single jump.
- Large deviations # edges \(E_n\) (vdHvdHKMM '24)
\(\mathbb{P}(E_n>\mathbb{E}[E_n] +n\nu)=^\ast (1+o(1))c_\nu n^{-(\tau-2)\lceil\nu\rceil}\). - Inhomogeneous random graphs (J., Zwart '24+)
\(\mathbb{P}\big(|\mathcal{C}_n^{(1)}|/n>\rho\big)=^\ast (1+o(1))C_\rho n^{-(\tau-2)\lceil \widetilde h(\rho)\rceil}.\)
\Theta(n^{-(\tau-2)\lceil h(\rho)\rceil})
$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{w_u^{1/d}+w_v^{1/d}}{\|x_u-x_v\|^d}\wedge 1\bigg)^{d\alpha}$$
Challenge: Capture dependency on percolation \(p\)
Power-law weights
Long-range percolation
Lower tail*:
If \(\mathbb{E}[\# \text{edges of length } n^{1/d}]\to\infty\)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)}\big)
\sim\exp\big(-\mathbb{E}[|\searrow\mathrm{VB}(n)|]\big)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)
\sim\exp\big(-n^{\frac{d-1}d\vee(2-\alpha)\vee(
(\tau-1)/\alpha-(\tau-2)}\big)
\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, \(\rho\in(\theta, 1)\)
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
\sim^\ast n^{-I(\rho)}
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\rho\big)
*at continuity points of \(I(\rho)\)
\ggg
\lll
\sim \exp(-n)
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)
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 \(\mathbb{E}[\# \text{edges of length } n^{1/d}]\to\infty\)
If \(\zeta>(d-1)/d\)
(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 \(d\ge 2\)
- Rate functions (lower tail)
- Central limit theorem
- \(\zeta\in\big[(d-1)/d, 1\big)\)
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;
Annals of Probability (2025). -
Large deviations of the giant in supercritical KSRGs;
With Júlia Komjáthy, Dieter Mitsche;
Probability Theory & Related Fields (to appear).
Puerto Sonoro
Cumbia, Ska, Funk, Reggea
Friday, 20:30 (Club 1019), €10.55
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
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)
(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}}
\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