Positive Rates Conjecture (PRC) Overview

• Positive Rates Conjecture (PRC) refers to distinct proposals in percolation theory on Cayley graphs and in one-dimensional probabilistic cellular automata regarding ergodicity.
• In the percolation context, PRC posits that p₍c₎(G)=1 if and only if G has a finite-index cyclic subgroup, with progress achieved for indicable groups using methods like EIT and evolving-sets.
• In the PCA setting, the conjecture that strict positive transition rates ensure ergodicity was refuted, leading to a nuanced theory with multiple phase transitions in invariant measures.

Searching arXiv for the relevant papers and related formulations of "Positive Rates Conjecture." The expression Positive Rates Conjecture (PRC) denotes different conjectures in distinct research areas. In the literature considered here, the two principal meanings are a percolation-theoretic conjecture of Benjamini and Schramm on critical Bernoulli bond-percolation on Cayley graphs of finitely generated groups, and an ergodicity conjecture for one-dimensional probabilistic cellular automata (PCA) with strictly positive transition rates. The first concerns whether $p_c(G)=1$ occurs exactly for groups with a finite-index cyclic subgroup; the second asserted that strict positivity of local update probabilities forces ergodicity. The percolation conjecture remains open in full generality but has been proved for broad classes of groups, while the PCA conjecture was refuted by Gács and has since acquired a more intricate phase-transition theory (Raoufi et al., 2016, Marsan et al., 4 Jul 2025).

1. Terminological scope and principal formulations

Within the sources summarized here, “PRC” is used for more than one problem, and the shared acronym does not indicate a common theorem schema. The principal uses are as follows.

Setting	Formulation	Status in the sources
Finitely generated groups and Bernoulli bond-percolation	$p_c(G)=1$ iff $G$ has a finite-index cyclic subgroup	Open in full generality; proved for indicable groups and reduced to hereditary just-infinite groups (Raoufi et al., 2016)
One-dimensional PCA	Every one-dimensional PCA with strictly positive rates is ergodic	Refuted by Gács; later work constructs two ergodicity phase transitions (Marsan et al., 4 Jul 2025)
Hilbert-space problem tied to $\zeta(s)$	All Cholesky coefficients $a_{n,k}=\langle f_n,e_k\rangle$ are positive	A distinct positivity conjecture, acronymally similar but not the same problem (Bellemare et al., 2020)
Short intervals and arithmetic progressions for primes	Positive-density occupation of intervals of length $\lambda\log X$ or residue classes modulo $q$	Presented as PRC-style conclusions rather than a standard “Positive Rates Conjecture” (Sardari, 2018)

The two canonical meanings differ sharply in ontology and methodology. In group percolation, the central object is the critical parameter $p_c$ for Bernoulli bond-percolation on a Cayley graph; in PCA, the central object is a Markov operator $\Phi$ on $M(A^{\mathbb Z})$ induced by a local stochastic rule with strictly positive outputs (Raoufi et al., 2016, Marsan et al., 4 Jul 2025).

2. The percolation-theoretic PRC on finitely generated groups

In its original formulation, the Positive Rates Conjecture of Benjamini and Schramm states:

$p_c(G)=1$0

Equivalently, any finitely generated group that is not virtually $p_c(G)=1$1 satisfies $p_c(G)=1$2 (Raoufi et al., 2016).

The relevant percolation parameter is defined on a Cayley graph. Given a finite symmetric generating set $p_c(G)=1$3 of a finitely generated group $p_c(G)=1$4, the right Cayley graph $p_c(G)=1$5 has vertex set $p_c(G)=1$6 and an edge $p_c(G)=1$7 whenever $p_c(G)=1$8. For Bernoulli bond percolation with retention parameter $p_c(G)=1$9, $G$0 denotes the product measure on edge states, and

$G$1

Since $G$2 depends only on the quasi-isometry class of $G$3, one writes $G$4 (Raoufi et al., 2016).

The paper “Indicable groups and $G$5” proves a major positive case of the conjecture. Its Theorem A states that if $G$6 is a finitely generated group admitting a non-trivial homomorphism

$G$7

then $G$8 unless $G$9 is virtually $\zeta(s)$0. Such groups are called indicable, equivalently groups with infinite abelianization (Raoufi et al., 2016).

The same work also provides a structural reduction. A finitely generated infinite group is just-infinite if every non-trivial normal subgroup has finite index, and hereditary just-infinite if every finite-index subgroup is itself just-infinite. Theorem B states that if the Positive Rates Conjecture holds for all finitely generated hereditary just-infinite groups of subexponential growth, then it holds for all finitely generated groups (Raoufi et al., 2016).

This reduction isolates the remaining obstruction. In the formulation given there, the only remaining potential counterexamples are finitely generated hereditary just-infinite groups of subexponential growth which do not virtually contain $\zeta(s)$1 (Raoufi et al., 2016).

3. Proof architecture for indicable groups and the reduction theorem

The proof strategy in the group-percolation setting combines probabilistic and algebraic inputs. The probabilistic core is the Exponential Intersection Tails (EIT) method. One constructs a probability measure $\zeta(s)$2 on infinite self-avoiding paths in a Cayley graph $\zeta(s)$3 of $\zeta(s)$4, and proves that there exists $\zeta(s)$5 such that for two independent samples $\zeta(s)$6,

$\zeta(s)$7

A theorem of Benjamini–Pemantle–Peres then implies that the existence of such an EIT measure yields $\zeta(s)$8; in the paper this appears as Theorem 6 in the form “EIT $\zeta(s)$9” (Raoufi et al., 2016).

The second ingredient is the evolving-sets method for time-inhomogeneous random walks. If $a_{n,k}=\langle f_n,e_k\rangle$0, then $a_{n,k}=\langle f_n,e_k\rangle$1, and the geometry is organized into a “fiber” $a_{n,k}=\langle f_n,e_k\rangle$2 over a “base” $a_{n,k}=\langle f_n,e_k\rangle$3. On $a_{n,k}=\langle f_n,e_k\rangle$4, one runs a time-dependent Markov chain whose transition matrices are Cayley-graph random-walk kernels conjugated by the motion in the base. When the Cayley graph of $a_{n,k}=\langle f_n,e_k\rangle$5 satisfies a uniform $a_{n,k}=\langle f_n,e_k\rangle$6-dimensional isoperimetric inequality with $a_{n,k}=\langle f_n,e_k\rangle$7, the evolving-sets technique yields a Gaussian upper bound on the diagonal heat kernel:

$a_{n,k}=\langle f_n,e_k\rangle$8

From this, Corollary 8 derives exponential tails for intersections of two independent copies of the walk, thereby feeding into the EIT criterion (Raoufi et al., 2016).

The required isoperimetry is obtained from growth. Theorem 10 states that if a Cayley graph of $a_{n,k}=\langle f_n,e_k\rangle$9 satisfies $\lambda\log X$0 with $\lambda\log X$1, then it has a $\lambda\log X$2-dimensional isoperimetric inequality. The paper further uses the fact that if $\lambda\log X$3 has strictly super-quadratic volume growth, then Gromov/Coulhon–Saloff-Coste gives an isoperimetric dimension $\lambda\log X$4. If $\lambda\log X$5 is virtually $\lambda\log X$6 or $\lambda\log X$7, one embeds a copy of $\lambda\log X$8 into $\lambda\log X$9, whence $q$0. When $q$1 has subexponential growth and $q$2 is nontrivial, Rosset’s theorem ensures that $q$3 is finitely generated, so this dichotomy applies (Raoufi et al., 2016).

The reduction to hereditary just-infinite groups uses the trichotomy of just-infinite groups into branch groups, groups containing a direct power of a hereditary just-infinite group, and hereditary just-infinite groups themselves. In the first two cases, the argument yields embeddings of high-dimensional lattices, hence $q$4. This is the mechanism behind the general reduction theorem (Raoufi et al., 2016).

4. The PCA PRC: positive rates versus ergodicity in one dimension

In the PCA setting, the Positive Rates Conjecture concerns a one-dimensional lattice $q$5 with finite alphabet $q$6. A probabilistic cellular automaton of “positive rates” is specified by a radius $q$7 and a local stochastic rule

$q$8

such that

$q$9

This induces a Markov operator $p_c$0 on $p_c$1 by

$p_c$2

The operator $p_c$3 is called ergodic if there exists a unique invariant measure $p_c$4 with $p_c$5, and for every initial $p_c$6,

$p_c$7

in the weak sense. The Positive Rates Conjecture asserted that every one-dimensional PCA with strictly positive rates must be ergodic (Marsan et al., 4 Jul 2025).

This conjecture was refuted by Gács. In the 1986 construction, later expanded in 2001 and 2024, Gács built a one-dimensional PCA on a huge finite alphabet $p_c$8 and radius $p_c$9 which admits two distinct invariant measures and therefore fails to be ergodic when the uniform noise rate $\Phi$0 is below a small critical threshold $\Phi$1. The local rule simulates an ever-growing hierarchy of encoded copies of itself, so that small islands of “errors” proliferate rather than dying out. In the formulation summarized here, the non-ergodic regime $\Phi$2 yields at least two distinct extremal invariant measures, supported on “error-free” and “error-rich” phases (Marsan et al., 4 Jul 2025).

The same source emphasizes the asymmetry between low and high noise. It is classical, via coupling-from-the-past and oriented percolation arguments, that any one-dimensional PCA with finite radius becomes ergodic as the noise rate $\Phi$3. The cited work of Vasilyev (1978), Gray (1982), and Marsan–Sablik–Törmä (2019) shows that there exists $\Phi$4, depending only on the radius, such that for all $\Phi$5 the PCA is uniformly ergodic. Conversely, all known counterexamples to the original PRC are non-ergodic only for sufficiently small noise, and their intermediate-noise behavior had remained unclear (Marsan et al., 4 Jul 2025).

5. Two ergodicity phase transitions and the revised PCA landscape

A further development is the construction by Hugo Marsan, Mathieu Sablik, and Ilkka Törmä of a radius-$\Phi$6 PCA

$\Phi$7

whose uniform $\Phi$8-noise perturbation has three regimes:

1. for $\Phi$9 it is ergodic;
2. at $M(A^{\mathbb Z})$0 it is non-ergodic;
3. for $M(A^{\mathbb Z})$1 (close to $M(A^{\mathbb Z})$2) it is again ergodic,

with $M(A^{\mathbb Z})$3 (Marsan et al., 4 Jul 2025).

The construction superposes two layers. The $M(A^{\mathbb Z})$4-layer is Gács’s non-ergodic cellular automaton $M(A^{\mathbb Z})$5. The $M(A^{\mathbb Z})$6-layer has alphabet

$M(A^{\mathbb Z})$7

Its noiseless update increments counters modulo $M(A^{\mathbb Z})$8 when no arrow arrives, sends marker arrows $M(A^{\mathbb Z})$9 rightwards with limited lifetime $p_c(G)=1$00, and synchronizes the counter value of cells an arrow passes through. The resulting mechanism ensures that at every $p_c(G)=1$01-th step each cell produces a $p_c(G)=1$02 on $p_c(G)=1$03, which projects a forced $p_c(G)=1$04 on the $p_c(G)=1$05-layer. The deterministic CA $p_c(G)=1$06 resets the $p_c(G)=1$07-symbol to $p_c(G)=1$08 whenever the $p_c(G)=1$09-layer at that site is $p_c(G)=1$10 or $p_c(G)=1$11 with $p_c(G)=1$12 (Marsan et al., 4 Jul 2025).

The noisy local rule is given by

$p_c(G)=1$13

where $p_c(G)=1$14. The three regimes are justified by different mechanisms. At very low noise, arrows in the $p_c(G)=1$15-layer last long enough to create macroscopic synchronized blocks of length $p_c(G)=1$16, and these blocks enforce a synchronous $p_c(G)=1$17 every $p_c(G)=1$18 steps on the $p_c(G)=1$19-layer, blocking information flow across them; a coupling-from-the-past argument then shows ergodicity. At an intermediate parameter, specifically $p_c(G)=1$20 with $p_c(G)=1$21 large, the $p_c(G)=1$22-noise rarely forms long arrows, so it acts combinatorially like an even smaller uniform noise on $p_c(G)=1$23, below Gács’s threshold $p_c(G)=1$24, allowing two invariant measures. At high noise, ergodicity follows from the general high-noise contraction argument of MST 2019 (Marsan et al., 4 Jul 2025).

This is identified in the source as the first example with more than one phase transition in ergodicity as the noise varies. It shows that positivity of rates alone neither forces ergodicity nor precludes multiple phase transitions. The open questions posed there include characterizing the possible shapes of the set $p_c(G)=1$25, asking whether infinitely many alternating ergodic and non-ergodic intervals can occur, and determining the intermediate-noise behavior of Gács’s original automaton (Marsan et al., 4 Jul 2025).

6. Acronymal overlap and adjacent uses in analytic number theory

The acronym “PRC” also appears in nearby but distinct number-theoretic contexts. One is the Positivity Conjecture associated with the Nyman–Báez-Duarte criterion for the Riemann Hypothesis. There one considers the step functions

$p_c(G)=1$26

the Gram matrix $p_c(G)=1$27, and its Cholesky factorization $p_c(G)=1$28. Writing

$p_c(G)=1$29

the conjecture is that $p_c(G)=1$30 for every $p_c(G)=1$31 and $p_c(G)=1$32. This is empirically supported up to $p_c(G)=1$33, but it is a positivity statement about Cholesky coefficients rather than a positive-rates conjecture in the percolation or PCA sense (Bellemare et al., 2020).

The same paper is embedded in the Nyman–Báez-Duarte framework, where

$p_c(G)=1$34

and

$p_c(G)=1$35

It further records the Mellin-transform identity

$p_c(G)=1$36

and the integral formula

$p_c(G)=1$37

but no implication between this positivity conjecture and the Riemann Hypothesis has been established (Bellemare et al., 2020).

A second adjacent usage arises in Sardari’s work on primes in short intervals and arithmetic progressions, which is presented as yielding Positive-Rates-Conjecture-style conclusions. For fixed $p_c(G)=1$38, if

$p_c(G)=1$39

then a positive proportion of intervals of length $p_c(G)=1$40 in $p_c(G)=1$41 contain at least one prime, with lower density

$p_c(G)=1$42

Similarly, for residue classes modulo $p_c(G)=1$43, if $p_c(G)=1$44, then a positive proportion of classes contain a prime $p_c(G)=1$45 (Sardari, 2018).

These number-theoretic examples do not define the standard PRC of either percolation theory or one-dimensional PCA. They nonetheless illustrate how the acronym or its “positive rates” rhetoric migrates across fields, typically to designate a positivity or positive-density phenomenon rather than a single unified conjectural program (Bellemare et al., 2020, Sardari, 2018).