Large deviations of the giant in spatial random graphs

Joost Jorritsma

joint with Júlia Komjáthy, Dieter Mitsche

Probability Seminar University of Oxford, November '24

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)

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}[|\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}[|\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 tail: a too small largest component

|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \theta:=\mathbb{P}^0\big(0\leftrightarrow\infty\big)

\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\vartheta- \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

\[\lim_{k\to\infty}\lim_{n\to\infty}\mathbb{P}^0(|\mathcal{C}_n(0)|>k, 0\notin\mathcal{C}_n^{(1)}) =0\]

$|\mathcal{C}_n^{(2)}|=\Theta(\mathrm{polylog}(n))$

Edge set $\mathcal{E}_\infty$

Long-range parameter $\alpha>1$,
Percolation $p\in(0,1]$

Connection probability

Vertex set $\mathcal{V}_\infty$

Locations: Poisson point process intensity $\beta>0$
Power-law i.i.d. weights $\tau>2$:
$\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$, $w\ge 1$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{w_u^{1/d}+ w_v^{1/d}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$

\mathbb{P}\big(|{\color{blue}\text{largest}}|/n<\vartheta- \varepsilon \big)\lesssim \exp\big(-n^\zeta\big)

Goal 2 Prove

|{\color{blue}\text{largest}}|/n \overset{\mathbb{P}}\longrightarrow \theta

\Longrightarrow

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}[|\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}[|\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}[|\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}[|\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 $\zeta>(d-1)/d$

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.

Kernel-based spatial random graphs

Vertex set $\mathcal{V}_\infty$

Spatial locations $x_v\in\mathbb{R}^d$: PPP(1),
i.i.d. weights $w_v\ge 1$: Pareto($\tau$),

Edge set $\mathcal{E}_\infty$

Symmetric kernel $\kappa(w_1, w_2)$,
Edge-density $\beta>0$,
Long-range parameter $\alpha>1$,

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 $x_v\in\mathbb{R}^d$: PPP(1), or $\mathbb{Z}^d$,
i.i.d. weights $w_v\ge 1$: $\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

Edge set $\mathcal{E}_\infty$

Symmetric kernel $\kappa(w_1, w_2)$,
Edge-density $\beta>0$,
Long-range parameter $\alpha>1$,

\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}

\sigma=\tau-1

\tau=2

\sigma=1

\sigma=\tau-2

\sigma=0

SFP/GIRG

Hyperbolic RG

Age-dep. RCM

Scale-free Gilbert

Long-range percolation

Random geom. graph

Nearest-neighbor percolation

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)

|{\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 $d=1$

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

\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}

\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

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 $w_v\ge 1$:
$\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

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)

\mathbb{P}\big(\nexists\text{ backbone}\big)

\lesssim (n/k)\exp\big(-k^{2-\alpha}\big)

\Bigg\}

\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)

Soft Poisson Boolean model

Long-range percolation

Bond percolation on $\mathbb{Z}^d$

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta-\varepsilon\big)

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta+\varepsilon\big)

\ggg

Lower tail:

- surface tension

- vertex boundary

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

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{\beta}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{\beta}{\|x_u-x_v\|^d}\phantom{\wedge 1}\bigg)^\alpha$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p\bigg(}\frac{1}{\|x_u-x_v\|^d}\phantom{\bigg)^\alpha\wedge 1}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p}\bigg(\frac{1}{\|x_u-x_v\|^d}\phantom{\wedge 1}\bigg)^\alpha$$

Soft Poisson Boolean model

Edge set $\mathcal{E}_\infty$

Long-range parameter $\alpha>1$,
Percolation $p\in(0,1]$

Connection probability

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{ \phantom{w_u^{1/d}+ w_v^{1/d}}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$

Vertex set $\mathcal{V}_\infty$

Spatial locations: Poisson point process intensity $\beta>0$
Power-law i.i.d. weights $w_v\ge 1$:
$\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{w_u^{1/d}+ w_v^{1/d}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{ {\color{darkred}w_u^{1/d}+w_v^{1/d}}}{\|x_u-x_v\|}\wedge 1\bigg)^{\alpha d}$$

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 $w_v\ge 1$:
$\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\frac{\beta\cdot (w_u\cdot w_v)}{\|x_u-x_v\|^d}\wedge 1\bigg)^\alpha$$

Geom. inhom. RG

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)

\lesssim\exp\big(-n^{{\color{darkgray}\frac{d-1}d\vee}(2-\alpha)}\big)

\sim\exp\big(-\mathbb{E}[|\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\frac{3-\tau}{2-(\tau-1)/\alpha}}\big)

\sim\exp\big(-\mathbb{E}[|\mathrm{VB}(n)|]\big)

Remainder: Lower tail LRP - Upper tail GIRG

*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)

{\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

Component of the origin ${\color{green}\mathcal{C}(0)}$: $$\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)= $$

Second-largest component: $$|{\color{red}\mathcal{C}_n^{(2)}}|=\Theta\big(\phantom{what}\big)$$

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

Component of the origin ${\color{green}\mathcal{C}(0)}$: $$\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)= $$

Second-largest component: $$|{\color{red}\mathcal{C}_n^{(2)}}|=\Theta\big(\phantom{what}\big)$$

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 ($d=1$)

Component of the origin ${\color{green}\mathcal{C}(0)}$: $$\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)= $$

Second-largest component: $$|{\color{red}\mathcal{C}_n^{(2)}}|=\Theta\big(\phantom{what}\big)$$

(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}}

\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)

|{\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

Largest component in spatial random graphs

By joostjor

Largest component in spatial random graphs

Large deviations of the giant in spatial random graphs

Supercritical Erdős–Rényi random graph \(G(n, \lambda/n)\)

Supercritical bond percolation on \(\mathbb{Z}^d\)

Soft Poisson Boolean model

Soft Poisson Boolean model

Soft Poisson Boolean model

Main result: large deviations for the giant component

Lower tail: a too small largest component

Goal 2 Prove

Main result: large deviations for the giant component

Upper tail: polynomial decay when \(\tau<\infty\)

Main result: large deviations for the giant component

Supercritical bond percolation on \(\mathbb{Z}^d\)

One decay exponent governing cluster sizes

Thank you!

Kernel-based spatial random graphs

The interpolating kernel

Lower bounds: cluster-size decay

Lower bounds: cluster-size decay

Lower bounds: large deviations

Lower bounds: \(|\mathcal{C}_n^{(2)}|\)

Kernel-based spatial random graphs

Upper bound: \(|\mathcal{C}_n^{(2)}|\)

Long-range percolation

Soft Poisson Boolean model

Kernel-based spatial random graphs

Remainder: Lower tail LRP - Upper tail GIRG

Upper bounds

Challenge: Delocalized components

Components in supercritical graphs

Components in supercritical graphs

Previous results

Conjecture: \(\exists \zeta\in \big[\tfrac{d-1}{d},1\big)\):

Lower bounds

Upper bounds

Upper bounds

What about other value \(\zeta_\ast\)?

Components in supercritical graphs

(Second-)largest component in supercritical spatial random graphs

Lower bounds

Upper bounds

Lower bounds

Upper bounds

Lower bounds

Upper bounds

Lower bounds

Upper bounds

Largest component in spatial random graphs

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