Largest component in spatial inhomogeneous random graphs

Large deviations of the giant in spatial random graphs

Joost Jorritsma

joint with Júlia Komjáthy, Dieter Mitsche

One World Probability Seminar, June '26

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)

[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)

Geom. Inhom. Random Graph

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

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

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

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

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

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

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

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

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

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

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

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)

\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\frac{3-\tau}{2-(\tau-1)/\alpha}}\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(\rho)}

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

*at continuity points of $I(\rho)$

\ggg

\lll

\sim \exp(-n)

Biskup ['04]

\mathbb{P}(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon)

\lesssim\exp\big(-n^{2-\alpha-o(1)}\big)

BFKRZ['23], $d,p\!=\!1, \alpha\!=\!\infty$

\mathbb{P}(|{\color{blue}\mathcal{C}_n^{(1)}}|/n\le \varepsilon)

\lesssim\exp\big(-n^{(3-\tau)/2}\big)

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)

\mathbb{P}\big(|{\color{green}\mathcal{C}(0)}|> 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}}|> 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 \theta:=\mathbb{P}\big({\color{green}0}\leftrightarrow\infty\big)

{\text{\color{blue}Lower tail large deviations}}

Truncation

Local convergence

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)

\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 most }\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)

[O'Connell '98; Andreis, König, A. Patterson '21]

\Bigg\}

n^{\alpha-1}

$m_n\sim n^{2-\alpha}$ boxes

\mathbb{E}\Big[\big|\big\{v: v \text{ has edge of length }\Omega(n^{1/d})\big\}\big|\Big] \sim n^{2-\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)

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)

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})

*at continuity points of $\lceil h(\rho)\rceil$ and $\lceil\nu\rceil$

H_{\mathcal{C}}(z):=\mathbb{E}\big[z^{|{\color{green}\mathcal{C}(0)}|}\big],

h(\rho):=\frac{|\log H^{(-1)}(1-\rho)|}{|\log (1-p)|}

$\lceil h(\rho)\rceil$:

# hubs ($w_v=\Theta(n)$) required to increase $|{\color{blue}\mathcal{C}_n^{(1)}}|$ from $\approx\theta n$ to $>\rho n$

\mathbb{P}(h \text{ vertices of weight }\Theta(n))

=\Theta\Big(\mathbb{P}\big(\mathrm{Bin}(n, \mathbb{P}(w_v\ge n))=\lceil h(\rho)\rceil\big)\Big)

=\Theta\Big(\mathbb{P}\big(\mathrm{Bin}(n, n^{-(\tau-1)})=\lceil h(\rho)\rceil\big)\Big)

= \Theta(n^{-(\tau-2)\lceil h(\rho)\rceil})

Intuition for the probability:

Connection probability

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

Upper tail: polynomial decay when $\tau<\infty$

H_{\mathcal{C}}(z):=\mathbb{E}\big[z^{|{\color{green}\mathcal{C}(0)}|}\big],

h(\rho):=\frac{|\log H^{(-1)}(1-\rho)|}{|\log (1-p)|}

$\lceil h(\rho)\rceil$:

# hubs ($w_v=\Theta(n)$) required to increase $|{\color{blue}\mathcal{C}_n^{(1)}}|$ from $\approx\theta n$ to $>\rho n$

\frac{|\{v\in \mathcal{V}_n: |\mathcal{C}_{\text{without-hubs}}(v)|=\ell\}|}{n}

Intuition

\frac{|\{\text{components of size } \ell\}|}n\overset{\mathbb{P}}\longrightarrow\frac{\mathbb{P}(|\mathcal{C}(0)|=\ell)}{\ell}

\mathbb{P}\big(\mathcal{C}\nsim \text{hubs} \, \big|\, |\mathcal{C}|=\ell, h\text{ hubs}\big)=

n-|\mathcal{C}_n^{(1)}| \approx

|\{\text{components of size } \ell\}|\approx (n/\ell)\cdot\mathbb{P}(|\mathcal{C}(0)|=\ell)

1-|\mathcal{C}_n^{(1)}|/n

\sum_{\ell\ge 1}\sum_{\mathcal{C}\in \mathcal{G}_{\text{without-hubs}}:|\mathcal{C}|=\ell}\ell (1-p)^{\ell h}

\sum_{\ell\ge 1}\sum_{\mathcal{C}\in \mathcal{G}_{\text{without-hubs}}:|\mathcal{C}|=\ell}\ell \mathbb{1}\big\{\mathcal{C}\nsim\mathrm{hubs}\big\}

(1-p)^{hl}

\overset{\mathbb{P}}\longrightarrow \mathbb{P}(|\mathcal{C}(0)|=\ell)

=H_{\mathcal{C}}(1-p)

\approx \sum_{\ell\ge 1} (1-p)^{\ell h}\mathbb{P}(|\mathcal{C}(0)|=\ell)

Connection probability

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

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)

\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\frac{3-\tau}{2-(\tau-1)/\alpha}}\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)

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;
Preprint arXiv: 2303.00712.
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

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)

{\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

Large deviations of the giant in spatial random graphs

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

Long-range percolation

Kernel-based spatial random graphs

Kernel-based spatial random graphs

Kernel-based spatial random graphs

Main result: large deviations for the giant component

Remainder: Lower tail LRP - Upper tail GIRG

Upper bounds

Lower tail of large deviations

Lower tail of large deviations

Main result: large deviations for the giant component

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

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

Main result: large deviations for the giant component

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

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

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