Nonamenable percolation and the two-point operator

Source and verification

This article explains family 214 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

On a graph with strong expansion, an infinite percolation cluster can exist without being unique. Family 214 approaches this nonuniqueness phase through an operator built from connection probabilities. Boundedness on square-summable functions is the bridge from graph expansion to critical percolation estimates.

🏷️ Preliminaries: expansion and two thresholds

For a finite vertex set $K$, let ${\partial }_{V}K$ be its external vertex boundary. A bounded-degree graph is nonamenable when

$$h=\underset{0<|K|<\infty }{\inf }\frac{|{\partial }_{V}K|}{|K|}>0.$$

Thus every finite set has a boundary proportional to its volume. Assume throughout that the graph is infinite, connected, locally finite, and quasi-transitive, meaning that there are finitely many vertex types under graph symmetries.

For independent bond percolation, $p_c$ is the threshold for existence of an infinite cluster. The uniqueness threshold $p_u$ is the infimum of parameters at which there is almost surely exactly one infinite cluster. The inequality $p_c<p_u$ gives a phase with infinitely many infinite clusters.

Introduce the two-point kernel and its operator on counting-measure ${\ell }^2(V)$:

$$\begin{array}{rl}{T}_p(x,y)& ={\mathbb{P}}_p(x\leftrightarrow y),\\ (T_{p}f)(x)& =\sum _y{T}_p(x,y)f(y).\end{array}$$

Initially this formula can be applied to finitely supported functions. The question is whether it extends to a bounded operator. Let $p_{2\to 2}$ be the supremum of parameters for which it does.

🏷️ The manuscript’s operator claim

Nonamenable critical bound

The manuscript claims

$$\begin{array}{c}\underset{p<p_c}{\sup }\Vert T_p{\Vert }_{2\to 2}=\Vert T_{p_c}{\Vert }_{2\to 2}<\infty ,\\ p_c<p_{2\to 2}\le p_u.\end{array}$$

It also states the corresponding nonuniqueness result, including simultaneous parameter control on compact subintervals of the nonuniqueness phase. (OpenAI, 2026)

The strict gap beyond $p_c$ is a separate part of the argument. A finite norm at one parameter does not automatically make the set of boundedness parameters open.

🏷️ Why finite susceptibility is a different condition

The susceptibility is the row sum

$${\chi }_p(x)=\sum _y{T}_p(x,y)={\mathbb{E}}_p|C_x|.$$

It tests the operator on the constant function, which is not in ${\ell }^2$ on an infinite graph. Thus a bounded ${\ell }^2$ operator can coexist with infinite susceptibility.

For the $d$-regular tree, $d\ge 3$, the unique path between vertices gives $T_p(x,y)=p^{\mathrm{dist}(x,y)}$. There are $d(d-1{)}^{r-1}$ vertices at distance $r\ge 1$, so

$${\chi }_p(o)=1+\sum _{r\ge 1}d(d-1{)}^{r-1}{p}^r.$$

This diverges at $p_c=1/(d-1)$. The tree illustrates why an estimate phrased in terms of the operator norm can remain meaningful at criticality even when an expected cluster size does not.

A second elementary consequence is a triangle estimate. Nonnegativity and finite-volume approximation give

$$\begin{array}{rl}& \sum _{x,y}{T}_p(o,x)T_p(x,y)T_p(y,o)\\ & =⟨{\delta }_o,T_p^3{\delta }_o⟩\le \Vert T_p{\Vert }_{2\to 2}^3.\end{array}$$

The left side is a central quantity in mean-field percolation theory. This calculation explains the strength of the operator statement without establishing it.

🏷️ The proposed proof mechanism

The argument first derives weak but useful critical cluster moments. A map that changes the endpoint of a finite critical cluster disperses possible images, while controlling the probability cost of the change. If the cluster tail were too heavy, the number of resulting images would contradict this cost bound. The manuscript obtains finite moments of order below $1/2$ and near-critical susceptibility bounds with arbitrarily small power losses.

It then penalizes long pivotal corridors: a pivotal connection is one whose removal destroys the tested connection. Nonamenability gives a random-walk spectral radius below one, so movement through long corridors disperses mass efficiently. The modified connection kernel can have much smaller row sums while remaining close to the original kernel in the operator comparison relevant to the proof.

If the critical operator norm were unbounded, these estimates would force incompatible upper and lower bounds for a penalized kernel. Finally, a controlled increase of the edge parameter extends boundedness beyond criticality. The distinction between row mass and square-summable operator mass is used throughout; replacing one by the other would lose the argument.

🏷️ Key steps and proof outlines

The contradiction compares two effects of the same deformation: its operator norm barely changes, while its row mass decreases too quickly. “Deformation by pivotal corridors” constructs the kernel; “Critical boundedness and the nonuniqueness interval” specifies the scales that make the two estimates incompatible (OpenAI, 2026).

Sprinkling gives an explicit boundedness criterion

Let $A$ be the adjacency operator and let $d$ bound the degrees. Independently add edges with probability $a$ to a configuration of parameter $p$; their union has parameter $p+(1-p)a$. Split a simple connecting path at its added edges. The intervening original connections have disjoint witnesses, so the disjoint-occurrence inequality bounds their probability by the product of their connection probabilities. Summing intermediate vertices gives the entrywise estimate

$$T_{p+(1-p)a}\le \sum _{j\ge 0}{a}^j{T}_p(A{T}_p{)}^j.$$

Since $\Vert A\Vert \le d$, this series converges in operator norm if $ad\Vert T_p\Vert <1$. This proves the openness statement needed after critical boundedness. It also gives a lower bound under the contrary hypothesis: if $T_{p_c}$ were unbounded, with $\delta =p_c-p$ and $m_0=\Vert T_p\Vert$,

$$m_0\ge \frac{1-p}{d\delta }.$$

Otherwise sprinkling with $a=\delta /(1-p)$ would already bound the critical operator.

Weighting pivotal corridors by their probability cost

Take a symmetric lazy walk kernel $P$ with norm $\rho <1$, supplied by nonamenability. A raw $k$-step walk $w$ has probability $\mu (w)$. Erase its loops chronologically, obtaining a simple path ${\gamma }_w$ of length ${\ell }_w$. Retain only raw walks whose loop-erased path has ${\ell }_w\ge 3$. Its blockers are the edges incident to internal path vertices but not on the path; let their number be $b_w$.

A template is counted for a connection $x\leftrightarrow y$ if its path edges are all open and pivotal and all blockers are closed. Give it weight

$$c(w)=\mu (w)p^{-{\ell }_w}(1-p{)}^{-b_w}.$$

The reciprocal factors compensate for the probability of forcing that corridor open and its blockers closed. Distinct raw walks remain distinct templates even when their loop erasures agree. Let $N_{xy}$ be the total weight of counted templates and define

$$T_t(x,y)={\mathbb{E}}_p[1_{x\leftrightarrow y}{e}^{-t{N}_{xy}}].$$

Opening edges cannot increase the penalty on an existing connection: a newly counted pivotal template would already have been present on every old connecting path. This gives monotonicity needed for domain comparisons. Subcritical cluster moments dominate the derivatives, justifying differentiation under sums and expectations.

Write $m_t=\Vert T_t\Vert$, and let ${\chi }_t$ be the row sum averaged over the finitely many vertex types with the source’s fixed positive type weights. The construction gives an operator estimate of the form

$$m_t\ge m_0(1-2t{m}_0{\rho }^k).$$

The factor ${\rho }^k$ records dispersion of the raw walk. A separate path-selection argument finds two escape paths from a positive fraction of templates. Their lengths grow only logarithmically in the relevant row-to-norm ratio. Splicing these paths to connections outside the corridor yields a lower bound for $-{\chi }_t^{\prime }$ proportional to ${\chi }_t^2$, up to an error controlled by ${\rho }^k$ and a triangle term.

The scales force incompatible bounds

Set $S=\log (1/\delta )$, ${\chi }_0^{+}={\sup }_x{\sum }_y{T}_p(x,y)$, and $R={\chi }_0^{+}/m_0\ge 1$. Let $a_3$ be the type-averaged diagonal of $T_p^3$. The earlier moment and amplification estimates give

$$\log R=o(S),\log a_3=o(S).$$

Choose

$$\begin{array}{rl}k& =\lceil K_{\ast }[\log (2R)+\log (2+a_3)+\log S]\rceil ,\\ B_{\ast }& =(kR{)}^{C_2+2},t_{\ast }=B_{\ast }/m_0.\end{array}$$

Here $C_2$ comes from the escape-path probability and $K_{\ast }$ is fixed sufficiently large. Thus $k\to \infty$ but $k=o(S)$. The exponential decay of ${\rho }^k$ beats all displayed powers of $R,k,a_3$, while the exponential lower bound $m_0\gg e^S$ absorbs the corridor’s $e^{O(k)}$ local cost.

Consequently throughout $[0,t_{\ast }]$, the norm remains at least $m_0/2$, and row-to-norm comparison gives ${\chi }_t\ge m_0/(2c_{\sigma })$ for a fixed type constant $c_{\sigma }$. Meanwhile the escape estimate, after absorbing its triangle error, gives

$$-{\chi }_t^{\prime }\ge c(kR{)}^{-C_2}{\chi }_t^2.$$

Integrate the derivative of $1/{\chi }_t$ to obtain

$$\frac{1}{{\chi }_{t_{\ast }}}\ge \frac{c(kR{)}^2}{m_0},\frac{1}{{\chi }_{t_{\ast }}}\le \frac{2c_{\sigma }}{m_0}.$$

These inequalities contradict $kR\to \infty$. This is the exact role of the weak preliminary estimates: subpower growth is enough to separate the logarithmic corridor scale from $S$.

One coupling handles an interval of parameters

Once boundedness extends above $p_c$, choose $p_c<p_1<p_2$ with $T_{p_2}$ bounded. In the uniform-edge-label coupling, count the $p_2$-clusters containing an infinite $p_1$-cluster. This number is almost surely constant by ergodicity. It cannot be one: positive association and finitely many vertex types would give $T_{p_2}(x,y)\ge c>0$ uniformly, contradicting boundedness when tested on indicators of large finite sets. It cannot be a finite number greater than one either: opening a finite path merges two counted clusters with positive probability under an absolutely continuous finite modification, contradicting constancy.

Thus infinitely many distinct $p_2$-clusters contain infinite $p_1$-clusters. Those clusters remain infinite and distinct at every intermediate parameter on the same event. This avoids an invalid uncountable intersection of probability-one statements. The manuscript uses its separate nonunimodular input where the preceding type-averaged deformation argument does not apply directly.

See Also

• Family 213 concerns absence of an infinite cluster at criticality without assuming nonamenability.
• Family 219 uses the spectral geometry of regular trees for a different random operator problem.

🏷️ Manuscripts and verification scope

• Nonuniqueness of percolation on nonamenable quasi-transitive graphs — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes the strong critical operator bound and related nonuniqueness and critical-exponent results. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Nonuniqueness of percolation on nonamenable quasi-transitive graphs.