Percolation thresholds for critical bootstrap models with axis-contained update rules: sharp constants in two dimensions, coarse thresholds in higher dimensions

Bootstrap percolation with a general update family $\mathcal U$ (a monotone cellular automaton on $\mathbb Z^d$) has a critical probability $p_c$ whose asymptotics are governed by a geometric classification of $\mathcal U$ into subcritical, critical and supercritical families. In two dimensions the order of $p_c$ is known for every critical family by universality, but sharp thresholds (with leading constants) are known only for isolated examples: the 2-neighbour model, one unbalanced model without drift, the Duarte and modified Duarte models, and recently all symmetric voracious isotropic families. The paper 'The sharp threshold for the Duarte model' (Bollobás, Duminil-Copin, Morris, Smith, Ann. Probab. 2017) determined $p_c(\mathbb Z_n^2,\mathcal D)=(1/8+o(1))(\log\log n)^2/\log n$ and posed two open problems for the natural class of critical families whose update rules lie on the coordinate axes: Problem 7.2 asks for the sharp threshold for any such two-dimensional family, and Problem 7.4 asks, for each $d\ge 3$, for the coarse threshold (up to a constant factor) for any $d$-dimensional critical family with axis-contained rules. The later literature records that universality in all dimensions now fixes only the iterated-logarithmic scale $(1/\log^{(r-1)}n)^{\Theta(1)}$ for every critical family; that Blanquicett determined coarse thresholds for $r$-neighbour models with axial neighbourhoods in large parameter ranges; and that Duminil-Copin and Hartarsky (2024) proved the sharp metastability transition for symmetric voracious isotropic families while explicitly posing the general conjecture as open. Both problems remain open in their stated generality.

Background

Bootstrap percolation, also called $\mathcal U$-bootstrap percolation or a monotone cellular automaton, is an infection process on a lattice defined by a finite collection $\mathcal U=\{X_1,\dots,X_m\}$ of finite subsets of $\mathbb Z^d\setminus\{0\}$, called the update family, whose elements are update rules. Given a set $A$ of initially infected sites, at each discrete time step a healthy site $x$ becomes infected if some translate $x+X$ with $X\in\mathcal U$ is entirely infected, and infected sites never recover. The central quantity is the critical probability $p_c([n]^d,\mathcal U)=\inf\{p:\mathbb P_p(A\text{ percolates }[n]^d)>1/2\}$, where $A$ is $p$-random (each site infected independently with probability $p$) and percolation means that the closure of $A$, the set of eventually infected sites, is the whole box.

In two dimensions, Bollobás, Smith and Uzzell introduced a geometric classification that is now standard. For a direction $u$ on the unit circle $\mathbb S^1$, let $H_u=\{x\in\mathbb Z^2:\langle x,u\rangle<0\}$ be the discrete half-plane with outer normal $u$; $u$ is a stable direction if the closure of $H_u$ is $H_u$ itself. The family $\mathcal U$ is critical if some closed semicircle of $\mathbb S^1$ contains only finitely many stable directions while every open semicircle contains at least one; the remaining families are subcritical or supercritical, with critical probability respectively bounded away from zero or polynomial in $n$. Critical families are the delicate case, with critical probability poly-logarithmic in $n$. Two further invariants refine the critical class. The difficulty $\alpha(u)$ of a direction $u$ is the minimum number of infected sites that must be added to the half-plane $H_u$ for infinitely many further sites to become infected, and the difficulty of the family is $\alpha(\mathcal U)=\min_C\max_{u\in C}\alpha(u)$, where the minimum runs over open semicircles $C$ of $\mathbb S^1$. A critical family is unbalanced if some pair of opposite directions both have difficulty strictly larger than $\alpha(\mathcal U)$, and balanced otherwise. Unbalanced families are further divided into those with and without drift, according to whether, alongside a half-plane in a hardest direction, growth along the boundary line is biased to one side. The balanced families include the isotropic ones, in which there are only finitely many stable directions and every open semicircle contains a stable direction of maximal difficulty; for symmetric families the two notions coincide.

A threshold result is called sharp when the leading constant is pinned down (upper and lower bounds differing by a factor $1+o(1)$), and coarse when only the order of magnitude up to a constant factor is determined. The archetype of a sharp result is Holroyd's theorem for the 2-neighbour model (update family: all 2-element subsets of $\{\pm e_1,\pm e_2\}$): $p_c([n]^2)=\frac{\pi^2}{18\log n}+o(1/\log n)$. Balogh, Bollobás, Duminil-Copin and Morris later determined sharp thresholds for the $r$-neighbour model in every dimension (in which a site of $\mathbb Z^d$ becomes infected once at least $r$ of its $2d$ nearest neighbours are infected). At the coarse level, the two-dimensional universality theorem of Bollobás, Duminil-Copin, Morris and Smith states that every critical family of difficulty $\alpha$ satisfies $p_c([n]^2,\mathcal U)=\Theta((1/\log n)^{1/\alpha})$ if balanced and $p_c([n]^2,\mathcal U)=\Theta(((\log\log n)^2/\log n)^{1/\alpha})$ if unbalanced. The same program has recently been extended to all dimensions by Balister, Bollobás, Morris and Smith: every critical $d$-dimensional family satisfies $p_c(\mathbb Z_n^d,\mathcal U)=(1/\log^{(r-1)}n)^{\Theta(1)}$ for some $r\in\{2,\dots,d\}$, where $\log^{(k)}$ denotes the $k$-times iterated logarithm and criticality in $\mathbb Z^d$ is defined via the stable set of directions on the sphere $\mathbb S^{d-1}$.

Sharp thresholds, however, are known in only a handful of two-dimensional cases: the 2-neighbour model and its isotropic relatives (recently generalized by Duminil-Copin and Hartarsky to every symmetric voracious isotropic family, a helping set being a minimum-size set of sites which, added to a half-plane, causes unbounded further infection, and voracity the property that such helping sets next to a half-plane consume the whole adjacent line), a single unbalanced model without drift due to Duminil-Copin and van Enter (the anisotropic threshold model whose neighbourhood is $\{(\pm2,0),(\pm1,0),(0,\pm1)\}$, in which a site becomes infected once three of these six neighbours are infected), and the Duarte models. The Duarte model has update family $\mathcal D$ consisting of the 2-element subsets of $\{(-1,0),(0,1),(0,-1)\}$; it is critical, unbalanced, and has drift, and its two-rule subfamily is the modified Duarte model. Mountford determined its threshold up to a constant factor in 1995, and the source paper determined the sharp threshold

$$p_c(\mathbb Z_n^2,\mathcal D)=\Big(\frac18+o(1)\Big)\frac{(\log\log n)^2}{\log n},$$

resolving Mountford's question and giving the first sharp result for a model with drift; the proof required non-polygonal curved droplets and very sharp bounds on crossing probabilities. (The source paper works on the torus $\mathbb Z_n^2=\mathbb Z^2/n\mathbb Z^2$, on which the critical probability is defined verbatim as above, with percolation of the whole torus.)

An update family has its rules contained in the axes if every site of every rule lies on a coordinate axis (in two dimensions: $ab=0$ for every $(a,b)$ in every rule $X\in\mathcal U$). This class contains the 2-neighbour, Duarte, modified Duarte and Duminil-Copin–van Enter models, and is the natural next target of the program. In higher dimensions the landscape is much sparser: beyond the $r$-neighbour models, even coarse thresholds were, at the time of the source paper, known only for a family of symmetric three-dimensional threshold models with axis-contained rules studied by van Enter and Fey. In Section 7 of the Duarte paper the authors therefore posed the two problems that are the subject of this record.

Problem Statement

The problem is the pair of open problems posed in Section 7 of 'The sharp threshold for the Duarte model' (Bollobás, Duminil-Copin, Morris, Smith), which share one scientific objective: determine the percolation threshold asymptotics for critical bootstrap models whose update rules are contained in the coordinate axes.

Part 1 (two dimensions; Problem 7.2 of the source). Determine the sharp threshold, that is, upper and lower bounds on $p_c(\mathbb Z_n^2,\mathcal U)$ differing by a factor $1+o(1)$, including the leading constant, for any critical two-dimensional update family $\mathcal U$ whose update rules are contained in the axes: for every $X\in\mathcal U$ and every $(a,b)\in X$ one has $ab=0$. The class contains both balanced and unbalanced families. The solved benchmark cases inside the class are the 2-neighbour model ($(\pi^2/18+o(1))/\log n$), the Duminil-Copin–van Enter model and the Duarte and modified Duarte models ($(1/12+o(1))$, $(1/8+o(1))$ and $(1/4+o(1))$ times $(\log\log n)^2/\log n$); the problem asks for the analogous determination, with proof, for an arbitrary critical axis-contained family, establishing in particular that a limiting constant exists and identifying it. In the form conjectured by later literature (Conjecture 6.1 of Duminil-Copin–Hartarsky), for a family of difficulty $\alpha$ the infection time $\tau$ of the origin in the process on $\mathbb Z^2$ with $p$-random initial set (the first time at which the origin becomes infected) should satisfy $p^\alpha\log\tau/(\log(1/p))^\gamma\to\lambda(\mathcal U)$ in probability as $p\to0$, with $\gamma=0$ for balanced and $\gamma=2$ for unbalanced families, the box threshold being obtained by inversion; but the problem as stated by the source is to determine the sharp threshold, whatever its precise form.

Part 2 (higher dimensions; Problem 7.4 of the source). For each $d\ge3$, determine the coarse threshold, that is, the critical probability up to a constant factor, for any $d$-dimensional critical family whose update rules are contained in the axes. Here criticality is defined via the stable set $\mathbb S=\{u\in\mathbb S^{d-1}:[H_u^d]=H_u^d\}$ of directions whose discrete half-space is stable: the family is critical if the spherical measure of $\mathbb S\cap C$ is zero for some hemisphere $C\subset\mathbb S^{d-1}$ while $\mathbb S\cap C$ is nonempty for every open hemisphere; axis-containment means every site of every rule lies on some coordinate axis. The problem as posed is the general class, and the openness claimed here concerns that generality.

The task is to determine, for every family in the stated classes, the threshold at the requested precision (a sharp leading constant in two dimensions; two-sided constant-factor bounds in each dimension $d\ge3$), with proof. The two parts are linked as stated by the source: Part 2 is the higher-dimensional analogue of Part 1 at coarser precision, and both target the same class of axis-contained critical families, of which the 2-neighbour, Duarte and Duminil-Copin–van Enter models are the archetypal solved instances.

References

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