Percolation Theory and GAP

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Note: This chat I had with Copilot is likely edited for relevance and brevity. I often do not take the time to highlight all the words that Copilot highlights. Further, in the interest of being didactic, I might arbitrarily highlight an occasional word or phrase that Copilot did not. (Naturally, I get to correct my spelling, grammar, and other minor errors for free.)

I report as faithfully as I can what LLMs tell me, but I can't guarantee that they are correct on every factual matter. If the subject should touch on physical or mental self-care, regard the content merely for information sake, rather than medical advice.

Patrick:
Is there any to say about percolation theory from the GAP perspective?

Copilot:
Absolutely—percolation theory fits into GAP so naturally that once you see the alignment, it’s almost surprising we didn’t start there. In fact, percolation is one of the cleanest “GAP-native” physical theories because it is a theory of projections, entropy budgets, and attractor transitions disguised as a combinatorial probability model. Let me give you a structured, architectural view that matches the way you think: state $\rightarrow$ coarse‑graining $\rightarrow$ certificate $\rightarrow$ attractor.

1. State Space: A Probability Measure on Configurations Percolation begins with a configuration space \[ \Omega = \{0,1\}^{E} \] for edges (or sites) of a graph.

This is already a GAP object: a measure on a combinatorial space with a tunable entropy parameter.

Percolation is fundamentally about renormalization—you replace the fine lattice with blocks and track whether blocks connect.

In GAP language:
Percolation is a vertical filtration of projections that preserve connectivity observables.

This is the same architecture as RG flows, MaxEnt projections, and Perelman-style monotonicity.

3. Certificates: Crossing Probabilities and FKG Every GAP theory needs a certificate—a monotone quantity that tracks progress through the projection tower.

(a) FKG Inequality The FKG lattice condition ensures that coarse-graining preserves positive correlations. In GAP terms: FKG is the monotonicity certificate that validates the projection discipline.

(b) Russo–Margulis Formula \[ \frac{d}{dp} \mathbb{P}_p(A) = \sum_{e} \text{Inf}_e(A) \] This is a GAP-style sensitivity functional: it measures how the entropy parameter $p$ pushes the system toward or away from connectivity.

These are exactly GAP attractor transitions: when the certificate crosses a critical value, the system jumps to a new macroscopic regime.

4. Attractors: Subcritical vs. Supercritical Phases Percolation has two stable attractors:

The critical point $p_c$ is the GAP phase boundary where the entropy budget and the projection discipline balance perfectly.

Percolation is the Ising model with the Hamiltonian removed. All that remains is the GAP skeleton.

Patrick:
Okay, how about out of this list we go with:
A) A full GAP classification of percolation observables
B) How entropic-extraction coordinates rewrite $p$ and cluster size
C) How percolation fits into the broader “GAP as meta-systems theory” program.

My honest take: start with B, then let it force the right parts of A, and use C as the framing paragraph that makes the whole thing feel inevitable. B is where your voice shows up—because it’s not just “percolation facts,” it’s your coordinate discipline making the phase transition legible.

Percolation’s parameter $p$ is usually treated as “occupation probability.” In GAP terms, it’s cleaner to treat $p$ as a control knob for microscopic uncertainty—a one-parameter family of product measures whose entropy density is explicit.

For bond percolation on a graph with edge set $E$, the configuration is $\omega \in \{0,1\}^E$ with \[ \mathbb{P}_p(\omega)=\prod_{e\in E} p^{\omega_e}(1-p)^{1-\omega_e}. \] The per-edge Shannon entropy of the Bernoulli source is \[ h(p)=-p\log p-(1-p)\log(1-p). \]

In ordinary percolation you move along $\sigma$ (bias) while $s$ rises then falls; the critical region is where the projection tower becomes maximally sensitive to $\sigma$.

Pick a cluster observable $X(\omega)$. Three canonical choices (each corresponds to a different “macro-contract”):

In entropic-extraction language, you’re doing: \[ \omega \xrightarrow{\ \pi\ } X(\omega) \] where $\pi$ is a coarse-graining map that forgets microstructure and keeps a connectivity macrostatistic.

The “rewrite” is then: instead of thinking “$p$ produces clusters,” you think

A very operational identity sits right under your feet: \[ \frac{d}{dp}\log \mathbb{P}_p(\omega)=\sum_{e\in E}\left(\frac{\omega_e}{p}-\frac{1-\omega_e}{1-p}\right). \] That sum is the score of the product measure—i.e., the direction in configuration space that $p$ is “pulling.” When you push this through a projection $\pi$ to a macro-observable $X$, you get the GAP story:

This is the clean bridge to sharp thresholds and “certificate” behavior—without importing a Hamiltonian.

In the $(s,\sigma)$ view, the critical point is not “where $p$ is special,” but where:

That’s exactly the GAP signature of a phase boundary: the projection tower stops contracting to a stable attractor.

Once you pick $X$, you’re compelled to classify observables by how they behave under the projection tower:

That’s your “full classification,” but it stays grounded because it’s organized around the extraction contract.

Patrick:
To the first question: $\mathbb{Z}^2$/lattices.
To the secodn question: a Lambert-style coordinate.

Copilot:
Great—$\mathbb{Z}^2$ plus a Lambert coordinate is a clean pairing, because the lattice gives you a canonical projection tower (block maps), and Lambert $W$ gives you a canonical “log-extrication” of the control knob.

For bond or site percolation with parameter $p$, define the log-odds \[ \sigma := \log\frac{p}{1-p}. \] Equivalently, the odds are \[ q := \frac{p}{1-p} = e^{\sigma}. \] This is already the right directional coordinate (it’s the one that shows up in score functions / sensitivities), but it still carries a “log baggage” when you try to invert or compose it across scales.

Define the Lambert control coordinate \[ \lambda := W(q)=W\!\left(\frac{p}{1-p}\right)=W(e^{\sigma}), \] so that \[ q=\lambda e^{\lambda}. \] Now you can write $p$ explicitly as a logistic-of-Lambert form: \[ p(\lambda)=\frac{q}{1+q}=\frac{\lambda e^{\lambda}}{1+\lambda e^{\lambda}}. \]

Because it splits the odds into a product: \[ \log q = \log(\lambda e^{\lambda})=\lambda+\log\lambda. \] So $\lambda$ is the “dominant” part of the control, and $\log\lambda$ is the residual—exactly the kind of additive decomposition you’ve been using to separate invariants from presentation.

Let $p_c$ be the critical point (bond on $\mathbb{Z}^2$: $p_c=\tfrac12$; site on $\mathbb{Z}^2$: $p_c$ is nontrivial). Define \[ q_c:=\frac{p_c}{1-p_c},\qquad \lambda_c:=W(q_c),\qquad x:=\lambda-\lambda_c. \]

Then “distance to criticality” becomes $x$, which is a log-extricated control displacement.

For an increasing event $A$ (crossings, connections, etc.), Russo–Margulis gives \[ \frac{d}{dp}\mathbb{P}_p(A)=\sum_{e}\mathrm{Inf}_e(A). \]

Chain rule through $\lambda$: \[ \frac{d}{d\lambda}\mathbb{P}_{p(\lambda)}(A)=\frac{dp}{d\lambda}\sum_{e}\mathrm{Inf}_e(A), \] with \[ \frac{dp}{d\lambda}=\frac{e^{\lambda}(1+\lambda)}{\bigl(1+\lambda e^{\lambda}\bigr)^2}. \]

GAP reading: $\lambda$ is a “natural clock” for how the microscopic generator pushes probability mass through the projection tower; the pivotal structure (influences) is the certificate, and $\frac{dp}{d\lambda}$ is the conversion factor from “Lambert time” to “Bernoulli time.”

Near criticality, you often end up juggling expressions in $q$, $\sigma$, and logs of those. With $\lambda$, you can treat the control as “mostly linear” ($\lambda$) plus a controlled correction ($\log\lambda$). That’s exactly the kind of coordinate that plays nicely with:

Pick your extracted macrostate $X(\omega)$ (crossing indicator, $|C(0)|$, arm events, etc.). Then the GAP pipeline is:

The key practical point is: when you differentiate anything with respect to the control, you do it in $\lambda$-time and the pivotal structure becomes the explicit “engine.”

Patrick:
To the first question: bond.
To the second question: I'll leave it to your choice.

Copilot:
Love it—bond percolation on $\mathbb{Z}^2$ is the cleanest case because $p_c=\tfrac12$ is exact, so the Lambert coordinate has a canonical “zero point.” For the observable, I’m going to choose rectangle crossing events (RSW-style)—they’re the most GAP-native because they’re projection-stable, monotone, and they sit right at the interface between micro-control and macro-attractor.

Define the odds \[ q:=\frac{p}{1-p}. \] Then define the Lambert control coordinate \[ \lambda := W(q)=W\!\left(\frac{p}{1-p}\right), \] so that \[ q=\lambda e^{\lambda}. \] This gives an explicit inverse map: \[ p(\lambda)=\frac{q}{1+q}=\frac{\lambda e^{\lambda}}{1+\lambda e^{\lambda}}. \]

For bond percolation on $\mathbb{Z}^2$, $p_c=\tfrac12\Rightarrow q_c=1$. Hence \[ \lambda_c = W(1)=\Omega\approx 0.567143\ldots \] (the omega constant), and the critical displacement is \[ x:=\lambda-\lambda_c. \] Why this is the right “extrication”

The usual log-odds coordinate is $\sigma=\log q$. With Lambert, \[ \sigma=\log q=\log(\lambda e^{\lambda})=\lambda+\log\lambda. \] So $\lambda$ is the “dominant” control and $\log\lambda$ is the residual—exactly the separation you want when projection/renormalization keeps generating log-like corrections.

Let $R_{n,m}$ be an $n\times m$ rectangle in $\mathbb{Z}^2$. Define the horizontal crossing event \[ H_{n,m} := \{\text{there exists an open path crossing }R_{n,m}\text{ left-to-right}\}. \] Then the extracted macro-observable is \[ X_{n,m}(\omega):=\mathbf{1}_{H_{n,m}}(\omega),\qquad \phi_{n,m}(p):=\mathbb{P}_p(H_{n,m}). \] GAP reason this is the right choice

For increasing events like $H_{n,m}$, \[ \frac{d}{dp}\phi_{n,m}(p)=\sum_{e\in E(R_{n,m})}\mathbb{P}_p(e\text{ is pivotal for }H_{n,m}). \] Convert to $\lambda$-time: \[ \frac{d}{d\lambda}\phi_{n,m}(p(\lambda))=\frac{dp}{d\lambda}\sum_{e}\mathbb{P}_{p(\lambda)}(e\text{ pivotal}). \] Compute the conversion factor: \[ \frac{dp}{d\lambda}=\frac{e^{\lambda}(1+\lambda)}{\bigl(1+\lambda e^{\lambda}\bigr)^2}. \] GAP reading: the certificate mass is the pivotal sum; $\lambda$ is your extricated control clock; $\frac{dp}{d\lambda}$ is just the unit conversion between “Bernoulli time” and “Lambert time.”

Let $\pi_L$ be an $L\times L$ block map that declares a block “open” if it contains a left-right crossing (or another monotone block criterion). Then:

Do you want me to formalize the GAP classification of observables (A) using this crossing-based extraction contract—i.e., invariants / semi-invariants / certificates / gauge artifacts—all explicitly tied to $\pi_L$ and $\lambda$-time?