--- title: Positive Rates Conjecture (PRC) Overview url: https://www.emergentmind.com/topics/positive-rates-conjecture-prc type: topic --- # Positive Rates Conjecture (PRC) Overview 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 [1606.06139] [2507.03485]. ## 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 [1606.06139] | | One-dimensional PCA | Every one-dimensional PCA with strictly positive rates is ergodic | Refuted by Gács; later work constructs two ergodicity phase transitions [2507.03485] | | 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 [2011.02847] | | 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” [1802.08524] | 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 [1606.06139] [2507.03485]. ## 2. The percolation-theoretic PRC on finitely generated groups In its original formulation, the Positive Rates Conjecture of Benjamini and Schramm states: \[ p_c(G)=1 \quad\Longleftrightarrow\quad G \text{ has a finite-index cyclic subgroup}. \] Equivalently, any finitely generated group that is not virtually \(\mathbb Z\) satisfies \(p_c(G)<1\) [1606.06139]. The relevant percolation parameter is defined on a Cayley graph. Given a finite symmetric generating set \(S\) of a finitely generated group \(G\), the right Cayley graph \(\mathcal T=(V,E)\) has vertex set \(V=G\) and an edge \(\{x,y\}\) whenever \(x^{-1}y\in S\). For Bernoulli bond percolation with retention parameter \(p\), \(\mathbb P_p\) denotes the product measure on edge states, and \[ p_c(\mathcal T)=\inf\bigl\{\,p\in[0,1]:\mathbb P_p\bigl(1 \text{ lies in an infinite open cluster}\bigr)>0\bigr\}. \] Since \(p_c(\mathcal T)\) depends only on the quasi-isometry class of \(\mathcal T\), one writes \(p_c(G)\) [1606.06139]. The paper “Indicable groups and \(p_c<1\)” proves a major positive case of the conjecture. Its Theorem A states that if \(G\) is a finitely generated group admitting a non-trivial homomorphism \[ \chi:G\to (\mathbb R,+), \] then \(p_c(G)<1\) unless \(G\) is virtually \(\mathbb Z\). Such groups are called **indicable**, equivalently groups with infinite abelianization [1606.06139]. 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 [1606.06139]. 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 \(\mathbb Z\) [1606.06139]. ## 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 \(\nu\) on infinite self-avoiding paths in a Cayley graph \(\mathcal T\) of \(G\), and proves that there exists \(c>0\) such that for two independent samples \(\gamma,\gamma'\sim\nu\), \[ \mathbb P\bigl(|\gamma\cap\gamma'|\ge k\bigr)\le e^{-ck} \qquad\text{for all }k\ge 1. \] A theorem of Benjamini–Pemantle–Peres then implies that the existence of such an EIT measure yields \(p_c(G)<1\); in the paper this appears as Theorem 6 in the form “EIT \(\Rightarrow p_c<1\)” [1606.06139]. The second ingredient is the **evolving-sets** method for time-inhomogeneous random walks. If \(N=\ker\chi\triangleleft G\), then \(G/N\cong \mathbb Z\), and the geometry is organized into a “fiber” \(N\) over a “base” \(G/N\). On \(N\), 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 \(N\) satisfies a uniform \(d\)-dimensional isoperimetric inequality with \(d>2\), the evolving-sets technique yields a Gaussian upper bound on the diagonal heat kernel: \[ P[X_t=y\mid X_0=x]\le C\,(K\,t)^{-d/2}. \] From this, Corollary 8 derives exponential tails for intersections of two independent copies of the walk, thereby feeding into the EIT criterion [1606.06139]. The required isoperimetry is obtained from growth. Theorem 10 states that if a Cayley graph of \(N\) satisfies \(|B(x,r)|\gtrsim r^d\) with \(d>2\), then it has a \(d\)-dimensional isoperimetric inequality. The paper further uses the fact that if \(N\) has strictly super-quadratic volume growth, then Gromov/Coulhon–Saloff-Coste gives an isoperimetric dimension \(d>2\). If \(N\) is virtually \(\mathbb Z\) or \(\mathbb Z^2\), one embeds a copy of \(\mathbb Z^2\) into \(G\), whence \(p_c(G)\le p_c(\mathbb Z^2)<1\). When \(G\) has subexponential growth and \(\chi\) is nontrivial, Rosset’s theorem ensures that \(N\) is finitely generated, so this dichotomy applies [1606.06139]. 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 \(p_c<1\). This is the mechanism behind the general reduction theorem [1606.06139]. ## 4. The PCA PRC: positive rates versus ergodicity in one dimension In the PCA setting, the Positive Rates Conjecture concerns a one-dimensional lattice \(\mathbb Z\) with finite alphabet \(A\). A probabilistic cellular automaton of “positive rates” is specified by a radius \(r\) and a local stochastic rule \[ \phi:A^{2r+1}\times A\to [0,1] \] such that \[ \phi(a,b)>0 \qquad \text{for all } a\in A^{2r+1},\ b\in A. \] This induces a Markov operator \(\Phi\) on \(M(A^{\mathbb Z})\) by \[ (\Phi\mu)([x_1\cdots x_n]_{i\cdots i+n-1}) =\int \mu(dx)\prod_{j=i}^{i+n-1}\phi(x_{j-r\cdots j+r},x_j). \] The operator \(\Phi\) is called **ergodic** if there exists a unique invariant measure \(\pi\) with \(\Phi\pi=\pi\), and for every initial \(\mu\in M(A^{\mathbb Z})\), \[ \Phi^t\mu \Rightarrow \pi \qquad \text{as } t\to\infty \] in the weak sense. The Positive Rates Conjecture asserted that every one-dimensional PCA with strictly positive rates must be ergodic [2507.03485]. 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 \(G\) and radius \(1\) which admits two distinct invariant measures and therefore fails to be ergodic when the uniform noise rate \(\rho\) is below a small critical threshold \(\rho_c>0\). 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 \(\rho<\rho_c\) yields at least two distinct extremal invariant measures, supported on “error-free” and “error-rich” phases [2507.03485]. 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 \(\rho\to 1\). The cited work of Vasilyev (1978), Gray (1982), and Marsan–Sablik–Törmä (2019) shows that there exists \(\rho'<1\), depending only on the radius, such that for all \(\rho\ge \rho'\) 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 [2507.03485]. ## 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-\(1\) PCA \[ T_\rho:(G\times F)^{\mathbb Z}\to (G\times F)^{\mathbb Z} \] whose uniform \(\rho\)-noise perturbation has three regimes: 1. for \(0<\rho<\rho_1\) it is ergodic; 2. at \(\rho=\rho_2\) it is non-ergodic; 3. for \(\rho>\rho_3\) (close to \(1\)) it is again ergodic, with \(0<\rho_1<\rho_2<\rho_3<1\) [2507.03485]. The construction superposes two layers. The \(G\)-layer is Gács’s non-ergodic cellular automaton \(G:G^{\mathbb Z}\to G^{\mathbb Z}\). The \(F\)-layer has alphabet \[ F=\{0,1,\dots,k-1\}\cup\{\nearrow_s:0\le s2k\), and these blocks enforce a synchronous \(0_G\) every \(k\) steps on the \(G\)-layer, blocking information flow across them; a coupling-from-the-past argument then shows ergodicity. At an intermediate parameter, specifically \(\rho=1/\ln\ln\ln k\) with \(k\) large, the \(F\)-noise rarely forms long arrows, so it acts combinatorially like an even smaller uniform noise on \(G\), below Gács’s threshold \(\rho_c\), allowing two invariant measures. At high noise, ergodicity follows from the general high-noise contraction argument of MST 2019 [2507.03485]. 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 \(\{\rho:F_\rho\text{ is ergodic}\}\), asking whether infinitely many alternating ergodic and non-ergodic intervals can occur, and determining the intermediate-noise behavior of Gács’s original automaton [2507.03485]. ## 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 \[ f_k(x)=\frac1k\Bigl\lfloor\frac1x\Bigr\rfloor-\Bigl\lfloor\frac1{kx}\Bigr\rfloor, \] the Gram matrix \(P_{j,k}=\langle f_j,f_k\rangle\), and its Cholesky factorization \(P=LL^T\). Writing \[ a_{n,k}=L_{n,k}=\langle f_n,e_k\rangle, \] the conjecture is that \(a_{n,k}>0\) for every \(n\ge 2\) and \(2\le k\le n\). This is empirically supported up to \(n=50\,000\), but it is a positivity statement about Cholesky coefficients rather than a positive-rates conjecture in the percolation or PCA sense [2011.02847]. The same paper is embedded in the Nyman–Báez-Duarte framework, where \[ d_n^2=\min_{\lambda_2,\dots,\lambda_n}\int_0^1\Bigl(1-\sum_{k=2}^n\lambda_k f_k(x)\Bigr)^2dx, \] and \[ \lim_{n\to\infty} d_n=0 \quad\Longleftrightarrow\quad \text{Riemann Hypothesis}. \] It further records the Mellin-transform identity \[ \int_0^1f_k(x)x^{s-1}dx = k^{-1}-k^{-s}s\zeta(s) \qquad (\Re s>0,\ s\neq 1), \] and the integral formula \[ \langle f_j,f_k\rangle =\frac1{2\pi j\,k}\int_{-\infty}^{\infty} \bigl(j^{\tfrac12-it}-1\bigr)\bigl(k^{\tfrac12+it}-1\bigr) \frac{|\zeta(\tfrac12+it)|^2}{\tfrac14+t^2}\,dt, \] but no implication between this positivity conjecture and the Riemann Hypothesis has been established [2011.02847]. 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 \(\lambda>0\), if \[ I_n(\lambda,X)=\pi(n+\lambda\log X)-\pi(n), \] then a positive proportion of intervals of length \(\lambda\log X\) in \([X,2X]\) contain at least one prime, with lower density \[ c(\lambda)=\frac{\lambda}{4\lambda+1}+o(1). \] Similarly, for residue classes modulo \(q\), if \(X\ge \lambda\varphi(q)\log q\), then a positive proportion of classes contain a prime \(\le X\) [1802.08524]. 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 [2011.02847] [1802.08524]. Source: https://www.emergentmind.com/topics/positive-rates-conjecture-prc