---
title: Adelic Percolation Model
url: https://www.emergentmind.com/topics/adelic-percolation-model
type: topic
---

# Adelic Percolation Model

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The adelic percolation model is a long-range percolation construction over a global field in which each place \(v\) carries a local percolation model on the completion \(F_v\), and a global edge \(\{x,y\}\) is declared present only when the corresponding local edges survive at all relevant places. In the formulation developed in "Adelic Models of Percolation" [2508.07601], this mechanism recasts two familiar systems—long-range percolation on ordinary lattices in \(\mathbb{R}^n\) and long-range percolation on hierarchical lattices—within a single adelic framework. The central structural input is the product formula for global fields, which identifies the Archimedean or \(\infty\)-adic fiber with a product of non-Archimedean contributions and thereby relates Euclidean, toric, and hierarchical kernels [2508.07601].

## 1. Classical long-range models and the adelic viewpoint

For a lattice \(\Lambda \subset \mathbb{R}^n\), such as \(\mathbb{Z}^n\) or \(O_K\) in the Minkowski embedding of a number field, the long-range percolation model is the random graph on \(\Lambda\) in which each unordered pair \(\{x,y\}\subset \Lambda\), \(x\neq y\), is connected independently with probability
\[
P_{\beta,\alpha}(\{x,y\}) = 1 - \exp(-\beta \|x-y\|^{-(n+\alpha)}),
\]
with \(\alpha>0\) and inverse temperature \(\beta \ge 0\). The standard questions are the existence of an infinite cluster above a critical \(\beta_c\), the decay of connectivities at criticality, and the dependence on \(\alpha\) [2508.07601].

The hierarchical model is built on
\[
H_L^n := \bigoplus_{i=1}^{\infty} (\mathbb{Z}/L\mathbb{Z})^n,
\]
equipped with the ultrametric
\[
d(x,y)=L^{h(x,y)}, \qquad h(x,y)=\max\{i:x_i\neq y_i\}.
\]
Its long-range percolation law is
\[
P_{\beta,\alpha,L}(\{x,y\}) = 1-\exp\big(-\beta\, d(x,y)^{-(n+\alpha)}\big).
\]
Because \(\sum_x d(0,x)^{-(n+\alpha)}<\infty\), the model is known, as noted in the source through Hutchcroft, to be always one-dimensional in critical behaviour and to admit sharper estimates on critical two-point functions than on \(\mathbb{Z}^n\) [2508.07601].

The adelic viewpoint identifies each of these classical models as a distinguished fiber of a larger construction. In the hierarchical case, the \(\infty\)-adic completion of a function field reproduces the ultrametric model. In the lattice case, the Archimedean part of a number field produces a toric model on the Minkowski lattice, and a separate one-parameter deformation then connects that toric kernel to the usual Euclidean kernel. This places apparently different geometries into a common local-to-global scheme [2508.07601].

## 2. Power-mean deformation and the toric kernel

Before adelization, the construction introduces a one-parameter interpolation between the Euclidean-norm kernel and a toric-volume-form kernel. For \(x=(x_1,\dots,x_n)\in \mathbb{R}^n\), a probability vector \(\lambda=(\lambda_i)\in \Delta_n\), and \(t\in \mathbb{R}\cup\{\pm \infty\}\), the power mean is
\[
M_t(\lambda,x)=\left[\sum_{i=1}^n \lambda_i |x_i|^t\right]^{1/t},
\]
with limits
\[
M_{-\infty}=\min_i |x_i|,\qquad M_0=\prod_i |x_i|^{\lambda_i},\qquad M_{+\infty}=\max_i |x_i|.
\]
The associated kernel is
\[
J_{\alpha,t,\lambda}(x-y):=M_t(\lambda,x-y)^{-(n+\alpha)},
\]
and the corresponding power-mean percolation model on \(\Lambda\) is
\[
P_{\beta,\alpha,t,\lambda}(x,y)=1-\exp\big(-\beta J_{\alpha,t,\lambda}(x-y)\big)
\]
[2508.07601].

Two special cases organize the comparison. When \(t=2\) and \(\lambda_i=1/n\), one recovers the usual Euclidean model up to the constant \(n^{(n+\alpha)/2}\):
\[
J_{\alpha,2,1/n}(x-y)=n^{(n+\alpha)/2}\|x-y\|^{-(n+\alpha)}.
\]
When \(t=0\) and \(\lambda_i=1/n\),
\[
M_0(1/n,x)=(|x_1\cdots x_n|)^{1/n}, \qquad
J_{\alpha,0,1/n}(x-y)=|x_1\cdots x_n|^{-(1+\alpha/n)}.
\]
Geometrically, this is the toric-volume-form percolation. If \(\Lambda\subset (\mathbb{R}^*)^n\) is transverse to the coordinate hyperplanes, then
\[
J_{\alpha,\mathrm{toric}}(x-y)= (|x_1\cdots x_n|)^{-(1+\alpha/n)}
\]
is the \((1+\alpha/n)\)-power of the Haar measure volume form
\[
\frac{dx_1\cdots dx_n}{x_1\cdots x_n}
\]
[2508.07601].

The monotonicity of \(M_t\) in \(t\) implies that for \(t<t'\), with \(\lambda\) fixed, inclusion probabilities decrease as \(t\) increases. In particular, toric percolation at \(t=0\) is always more connected than the Euclidean case at \(t=2\) [2508.07601]. This monotone deformation is the Archimedean bridge in the overall theory: it does not arise from the function-field adelization, but it is what connects the number-field toric fiber to ordinary lattice long-range percolation.

## 3. Function-field adelic model and recovery of hierarchical percolation

Let \(K=\mathbb{F}_q(C)\) be the function field of a smooth projective curve \(C/\mathbb{F}_q\). Its places are the closed points \(x\in C\) together with a chosen point at \(\infty\). For \(x\neq \infty\), the completion \(K_x\) is isomorphic to \(\mathbb{F}_{q_x}((t))\), with valuation
\[
|\cdot|_x=q_x^{-\operatorname{ord}_x(\cdot)},
\]
while at \(\infty\),
\[
K_\infty=\mathbb{F}_q((t^{-1})), \qquad |\cdot|_\infty=q^{-\deg(\cdot)}.
\]
The adelic product formula is
\[
\prod_{v\in \mathrm{places}} |f|_v =1 \qquad \forall f\in K^*.
\]
This identity is the arithmetic mechanism that links the local kernels [2508.07601].

At the place \(\infty\), the completion model reproduces hierarchical percolation exactly. On the abelian group \(\mathbb{F}_q[t]\cong (\mathbb{F}_q)^{\oplus \infty}\),
\[
J_{\alpha,q,\infty}(f-g)=|f-g|_\infty^{-(1+\alpha)}
= q^{(1+\alpha)\deg(f-g)}
= d(f,g)^{-(1+\alpha)},
\]
with \(d\) the ultrametric \(L^n\) distance. Proposition 3.3 identifies
\[
(_{q(C),\infty}) \cong (H_q^1)
\]
[2508.07601]. Thus the hierarchical lattice is not merely analogous to an \(\infty\)-adic geometry; it is the \(\infty\)-adic completion model of the function field in the stated sense.

At each finite place \(x\), the local kernel is defined by
\[
J_{\alpha,q,x}(f-g)=|f-g|_x^{(1+\alpha)}.
\]
On the same vertex set \(\mathbb{F}_q[t]\), one has \(\sum_f J_{\alpha,q,x}(f)=\infty\), so each local non-Archimedean model has an infinite cluster for any \(\beta>0\), equivalently critical \(\beta_c=0\) [2508.07601].

The adelic product model fixes parameters \(\beta_x,\alpha_x\) at each finite place and declares \(\{f,g\}\) to be an adelic edge exactly when all local edges survive. Equivalently,
\[
P_{\mathrm{adelic}}(\{f,g\})=
\prod_{x\ \mathrm{finite}}
\big[1-\exp(-\beta_x J_{\alpha_x,q,x}(f-g))\big].
\]
Using the product formula
\[
\prod_x |f-g|_x = |f-g|_\infty^{-1},
\]
Theorem 3.18 shows that if
\[
\beta_x=\beta\cdot \deg(x), \qquad \alpha_x=\alpha
\quad \text{or} \quad
\alpha_x=\alpha+\log_q \deg(x),
\]
then for large \(f-g\) the adelic probability is squeezed between two hierarchical-model probabilities with effective inverse temperatures \(\beta_S,\beta'_S\) built from partial zeta-functions \(Z^{(S)}(C,\beta)\) [2508.07601]. Proposition 3.20 then deduces that the existence or non-existence of an infinite adelic cluster is controlled by the critical \(\beta_c\) of the hierarchical case together with the \(\mathbb{F}_q(C)\)-zeta-function.

## 4. Number-field adelic model and Minkowski toric percolation

For a number field \(K\) with ring of integers \(O_K\), let the places split into non-Archimedean \( \nu\in P_f\) and Archimedean \( \sigma \in P_\infty\). The global product formula is
\[
\prod_{\nu\in P_f\cup P_\infty} |x|_\nu =1
\qquad \forall x\in K^*.
\]
The finite-place local models are defined on the countable subset \(\Sigma_{K,\nu}\subset K_\nu\) consisting of elements admitting a terminating \(\nu\)-adic expansion. With \(K_\nu\) the \(p\)-adic completion, residue field \(\mathbb{F}_{q_\nu}\), and valuation \(|\cdot|_\nu=q_\nu^{-v_\nu(\cdot)}\), one sets
\[
J_{\alpha,\nu}(x-y)=|x-y|_\nu^{(1+\alpha)}.
\]
Again, \(\sum_x J_{\alpha,\nu}(x)=\infty\), so every such local model has \(\beta_c=0\) [2508.07601].

The Archimedean side is constructed by the Minkowski embedding
\[
O_K \hookrightarrow \mathbb{R}^r \times \mathbb{C}^s
\]
through all \(r\) real and \(s\) complex embeddings. For each embedding \(\sigma:K\hookrightarrow \mathbb{R}\) or \(\mathbb{C}\),
\[
J_{\alpha,\sigma}(x-y)=|\sigma(x)-\sigma(y)|^{-(1+\alpha)}.
\]
Taking the product over \(\sigma\in P_\infty\) gives the Archimedean adelic model. Proposition 4.16 states that this \(\infty\)-adic adelic model on \(O_K\) is exactly the toric percolation \( _{t=0}(O_K)\) on the Minkowski lattice, up to a uniform re-indexing of \(\beta\):
\[
P_{\beta,\alpha/n,P_\infty}(\{x,y\}) \;\asymp\;
P_{\beta^n,\alpha,(r,s)}^{\mathrm{toric}}(\{x,y\}).
\]
This is the number-field counterpart to the function-field identification of the hierarchical model as the \(\infty\)-fiber [2508.07601].

The finite-place adelic model chooses
\[
\beta_\nu=\beta\cdot f_\nu, \qquad \alpha_\nu=\alpha
\quad \text{or} \quad
\alpha_\nu=\alpha+\log_p f_\nu,
\]
where \(f_\nu=[\mathbb{F}_{q_\nu}:\mathbb{F}_p]\), and defines
\[
P_{\mathrm{finite\mbox{-}adelic}}(\{x,y\})=
\prod_{\nu\in P_f}
\big[1-\exp(-\beta_\nu J_{\alpha_\nu,\nu}(x-y))\big].
\]
By partial Euler-product arguments for the Dedekind zeta \(\zeta_K(s)\), Theorem 4.22 shows that for large \(\|x-y\|\) this probability is squeezed between two toric-model probabilities on \(O_K\) with effective temperatures built from \(\zeta_K(\beta)\) [2508.07601]. The finite-place product therefore reproduces, in scaling, the same toric model that appears as the Archimedean adelic fiber.

## 5. Bridge theorem, examples, and comparison structure

The overall synthesis is summarized in Theorem 5.1 as a commutative bridge diagram. On the function-field side, the \(\infty\)-fiber is the hierarchical model \( (H_q^1)=(_{q(C),\infty})\), while the finite-place adelic product is equivalent in scaling to that same hierarchical model by the product formula. On the number-field side, the \(\infty\)-fiber is toric percolation on the Minkowski lattice, and the finite-place adelic product again recovers that toric model in scaling. The bottom horizontal map is the power-mean family on \(O_K\), interpolating between toric percolation at \(t=0\) and usual lattice percolation at \(t=2\) [2508.07601].

| Setting | \(\infty\)-fiber | Finite-place comparison |
|---|---|---|
| Function field \(\mathbb{F}_q(C)\) | Hierarchical percolation on \((\mathbb{F}_q)^{\oplus\infty}\) | Squeezed between hierarchical probabilities via \(Z^{(S)}(C,\beta)\) |
| Number field \(K\) | Toric percolation on the Minkowski lattice \(O_K\) | Squeezed between toric probabilities via \(\zeta_K(\beta)\) |
| Archimedean deformation | \(t=0\) toric kernel | \(t=2\) usual Euclidean lattice kernel |

Two examples in the source make the bridge explicit. For \(\mathbb{F}_q(t)\), the ring \(\mathbb{F}_q[t]\) is the hierarchical lattice \((\mathbb{F}_q)^{\oplus \infty}\) at \(\infty\), while at each finite prime \((t-a)\) the local model is again hierarchical with ultrametric determined by the order at \(a\); the finite-adelic product formula \(\prod_a |f|_a=|f|_\infty^{-1}\) recovers the same ultrametric percolation in scaling [2508.07601]. For a cyclotomic field \(K=\mathbb{Q}(\zeta_n)\), the Minkowski lattice \(O_K\hookrightarrow \mathbb{R}^{\varphi(n)}\) yields a genuine \(\mathbb{Z}^n\)-lattice model at \(\infty\), and its toric percolation can be compared to the full adelic model over \(p\)-adic primes through \(\zeta_K(\beta)\).

## 6. Conceptual significance, thresholds, and relation to standard percolation

The reason for introducing adelic geometry is a local-to-global synthesis. Both hierarchical and lattice percolations appear as special Archimedean fibers of adelic constructions, while the non-Archimedean fibers are simpler ultrametric models with trivial critical thresholds. The global product formula then ties the \(\infty\)-adic and non-\(\infty\)-adic contributions together and forces their long-range behaviours to coincide, yielding direct comparisons between hierarchical and Euclidean percolation regimes without passing to fractal limits or rigorous renormalization [2508.07601].

A common misunderstanding would be to infer from the finite-place fact \(\beta_c=0\) that the global adelic model must therefore percolate trivially. The stated results are more precise. In the function-field setting, Proposition 3.20 says that existence or non-existence of an infinite adelic cluster is controlled by the critical \(\beta_c\) of the hierarchical case together with the \(\mathbb{F}_q(C)\)-zeta-function. In the number-field setting, Proposition 5.2 says that the squeeze estimates imply percolation for \(\beta\) above a threshold \(\sim \max\{1,\beta_c\cdot \zeta_K(\beta)\}\) and non-percolation for \(\beta\) below \(\min\{1,\beta_c\cdot \zeta_K(\beta)\}\) [2508.07601]. Thus the finite-place local models are individually supercritical for every positive \(\beta\), but the global adelic edge law is constrained by the product structure and the zeta-function bounds.

The framework also situates classical models within a broader family. Ordinary long-range percolation on \(\mathbb{Z}^n\) uses the Euclidean kernel \(\|x-y\|^{-(n+\alpha)}\), while classical nearest-neighbour percolation is the limit \(\alpha\to\infty\) [2508.07601]. The adelic construction adds the observation that these real-variable models possess adelic shadows whose non-Archimedean factors are independent ultrametric percolations, and whose Archimedean part appears as the fiber at \(\infty\). A plausible implication is that the main novelty is not a new universality class by itself, but an arithmetic comparison principle between already familiar long-range systems.

Source: https://www.emergentmind.com/topics/adelic-percolation-model