Adelic Percolation Model

• Adelic percolation is a unifying framework that integrates local percolation models over completions using the global product formula.
• It bridges classical long-range percolation on Euclidean lattices and hierarchical percolation by relating their geometry through both Archimedean and non-Archimedean fibers.
• A one-parameter power-mean deformation interpolates between Euclidean and toric kernels, offering new insights into percolation thresholds and connectivity transitions.

Searching arXiv for the cited paper and closely related work. 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" (Marcolli, 11 Aug 2025), 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 (Marcolli, 11 Aug 2025).

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 $\{x,y\}\subset \Lambda$, $F_v$0, is connected independently with probability

$F_v$1

with $F_v$2 and inverse temperature $F_v$3. The standard questions are the existence of an infinite cluster above a critical $F_v$4, the decay of connectivities at criticality, and the dependence on $F_v$5 (Marcolli, 11 Aug 2025).

The hierarchical model is built on

$F_v$6

equipped with the ultrametric

$F_v$7

Its long-range percolation law is

$F_v$8

Because $F_v$9, 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 $\{x,y\}$0 (Marcolli, 11 Aug 2025).

The adelic viewpoint identifies each of these classical models as a distinguished fiber of a larger construction. In the hierarchical case, the $\{x,y\}$1-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 (Marcolli, 11 Aug 2025).

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,y\}$2, a probability vector $\{x,y\}$3, and $\{x,y\}$4, the power mean is

$\{x,y\}$5

with limits

$\{x,y\}$6

The associated kernel is

$\{x,y\}$7

and the corresponding power-mean percolation model on $\{x,y\}$8 is

$\{x,y\}$9

(Marcolli, 11 Aug 2025).

Two special cases organize the comparison. When $\mathbb{R}^n$0 and $\mathbb{R}^n$1, one recovers the usual Euclidean model up to the constant $\mathbb{R}^n$2: $\mathbb{R}^n$3 When $\mathbb{R}^n$4 and $\mathbb{R}^n$5,

$\mathbb{R}^n$6

Geometrically, this is the toric-volume-form percolation. If $\mathbb{R}^n$7 is transverse to the coordinate hyperplanes, then

$\mathbb{R}^n$8

is the $\mathbb{R}^n$9-power of the Haar measure volume form

$\infty$0

(Marcolli, 11 Aug 2025).

The monotonicity of $\infty$1 in $\infty$2 implies that for $\infty$3, with $\infty$4 fixed, inclusion probabilities decrease as $\infty$5 increases. In particular, toric percolation at $\infty$6 is always more connected than the Euclidean case at $\infty$7 (Marcolli, 11 Aug 2025). 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 $\infty$8 be the function field of a smooth projective curve $\infty$9. Its places are the closed points $\Lambda \subset \mathbb{R}^n$0 together with a chosen point at $\Lambda \subset \mathbb{R}^n$1. For $\Lambda \subset \mathbb{R}^n$2, the completion $\Lambda \subset \mathbb{R}^n$3 is isomorphic to $\Lambda \subset \mathbb{R}^n$4, with valuation

$\Lambda \subset \mathbb{R}^n$5

while at $\Lambda \subset \mathbb{R}^n$6,

$\Lambda \subset \mathbb{R}^n$7

The adelic product formula is

$\Lambda \subset \mathbb{R}^n$8

This identity is the arithmetic mechanism that links the local kernels (Marcolli, 11 Aug 2025).

At the place $\Lambda \subset \mathbb{R}^n$9, the completion model reproduces hierarchical percolation exactly. On the abelian group $\mathbb{Z}^n$0,

$\mathbb{Z}^n$1

with $\mathbb{Z}^n$2 the ultrametric $\mathbb{Z}^n$3 distance. Proposition 3.3 identifies

$\mathbb{Z}^n$4

(Marcolli, 11 Aug 2025). Thus the hierarchical lattice is not merely analogous to an $\mathbb{Z}^n$5-adic geometry; it is the $\mathbb{Z}^n$6-adic completion model of the function field in the stated sense.

At each finite place $\mathbb{Z}^n$7, the local kernel is defined by

$\mathbb{Z}^n$8

On the same vertex set $\mathbb{Z}^n$9, one has $O_K$0, so each local non-Archimedean model has an infinite cluster for any $O_K$1, equivalently critical $O_K$2 (Marcolli, 11 Aug 2025).

The adelic product model fixes parameters $O_K$3 at each finite place and declares $O_K$4 to be an adelic edge exactly when all local edges survive. Equivalently,

$O_K$5

Using the product formula

$O_K$6

Theorem 3.18 shows that if

$O_K$7

then for large $O_K$8 the adelic probability is squeezed between two hierarchical-model probabilities with effective inverse temperatures $O_K$9 built from partial zeta-functions $\Lambda$0 (Marcolli, 11 Aug 2025). Proposition 3.20 then deduces that the existence or non-existence of an infinite adelic cluster is controlled by the critical $\Lambda$1 of the hierarchical case together with the $\Lambda$2-zeta-function.

4. Number-field adelic model and Minkowski toric percolation

For a number field $\Lambda$3 with ring of integers $\Lambda$4, let the places split into non-Archimedean $\Lambda$5 and Archimedean $\Lambda$6. The global product formula is

$\Lambda$7

The finite-place local models are defined on the countable subset $\Lambda$8 consisting of elements admitting a terminating $\Lambda$9-adic expansion. With $\{x,y\}\subset \Lambda$0 the $\{x,y\}\subset \Lambda$1-adic completion, residue field $\{x,y\}\subset \Lambda$2, and valuation $\{x,y\}\subset \Lambda$3, one sets

$\{x,y\}\subset \Lambda$4

Again, $\{x,y\}\subset \Lambda$5, so every such local model has $\{x,y\}\subset \Lambda$6 (Marcolli, 11 Aug 2025).

The Archimedean side is constructed by the Minkowski embedding

$\{x,y\}\subset \Lambda$7

through all $\{x,y\}\subset \Lambda$8 real and $\{x,y\}\subset \Lambda$9 complex embeddings. For each embedding $F_v$00 or $F_v$01,

$F_v$02

Taking the product over $F_v$03 gives the Archimedean adelic model. Proposition 4.16 states that this $F_v$04-adic adelic model on $F_v$05 is exactly the toric percolation $F_v$06 on the Minkowski lattice, up to a uniform re-indexing of $F_v$07: $F_v$08 This is the number-field counterpart to the function-field identification of the hierarchical model as the $F_v$09-fiber (Marcolli, 11 Aug 2025).

The finite-place adelic model chooses

$F_v$10

where $F_v$11, and defines

$F_v$12

By partial Euler-product arguments for the Dedekind zeta $F_v$13, Theorem 4.22 shows that for large $F_v$14 this probability is squeezed between two toric-model probabilities on $F_v$15 with effective temperatures built from $F_v$16 (Marcolli, 11 Aug 2025). 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 $F_v$17-fiber is the hierarchical model $F_v$18, 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 $F_v$19-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 $F_v$20, interpolating between toric percolation at $F_v$21 and usual lattice percolation at $F_v$22 (Marcolli, 11 Aug 2025).

Setting	$F_v$23-fiber	Finite-place comparison
Function field $F_v$24	Hierarchical percolation on $F_v$25	Squeezed between hierarchical probabilities via $F_v$26
Number field $F_v$27	Toric percolation on the Minkowski lattice $F_v$28	Squeezed between toric probabilities via $F_v$29
Archimedean deformation	$F_v$30 toric kernel	$F_v$31 usual Euclidean lattice kernel

Two examples in the source make the bridge explicit. For $F_v$32, the ring $F_v$33 is the hierarchical lattice $F_v$34 at $F_v$35, while at each finite prime $F_v$36 the local model is again hierarchical with ultrametric determined by the order at $F_v$37; the finite-adelic product formula $F_v$38 recovers the same ultrametric percolation in scaling (Marcolli, 11 Aug 2025). For a cyclotomic field $F_v$39, the Minkowski lattice $F_v$40 yields a genuine $F_v$41-lattice model at $F_v$42, and its toric percolation can be compared to the full adelic model over $F_v$43-adic primes through $F_v$44.

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 $F_v$45-adic and non-$F_v$46-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 (Marcolli, 11 Aug 2025).

A common misunderstanding would be to infer from the finite-place fact $F_v$47 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 $F_v$48 of the hierarchical case together with the $F_v$49-zeta-function. In the number-field setting, Proposition 5.2 says that the squeeze estimates imply percolation for $F_v$50 above a threshold $F_v$51 and non-percolation for $F_v$52 below $F_v$53 (Marcolli, 11 Aug 2025). Thus the finite-place local models are individually supercritical for every positive $F_v$54, 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 $F_v$55 uses the Euclidean kernel $F_v$56, while classical nearest-neighbour percolation is the limit $F_v$57 (Marcolli, 11 Aug 2025). 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 $F_v$58. 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.