Fourth and higher-order correction terms for two-dimensional anisotropic bootstrap percolation, and the unproved $(1,b)$-family expansion

For the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood $\mathcal{N}_{(1,2)}=\{(\pm1,0),(0,\pm1),(\pm2,0)\}$ and threshold $r=3$, the critical probability is known to three asymptotic terms: $p_c([L]^2,\mathcal{N}_{(1,2)},3)=\frac{\log\log L}{12\log L}(\log\log L-4\log\log\log L+2\log\frac{9e}{2}\pm o(1))$ (Duminil-Copin, van Enter, Hulshof, 2018). The proof determines only two leading terms of the large-deviation probability that a critical droplet is internally filled, leaving a $o(\frac{1}{p}\log\frac{1}{p})$ remainder whose sharp order is undetermined; the authors suggest it may be as large as $O(\frac{1}{p}\log\log\frac{1}{p})$ and introduce a hypothetical next constant $C_3$ only conditionally. The same paper states a three-term expansion for the whole $(1,b)$-family, with an explicit constant $C(b)$, but withholds the proof as lengthy. This problem asks for the next correction terms: the sharp remainder in the droplet expansion (including the entropic-fluctuation contribution around the optimal growth trajectory), the induced fourth and further terms of $p_c$ for the $(1,2)$-model, and a complete rigorous proof of the $(1,b)$-family statement.

Background

Bootstrap percolation is a family of monotone cellular automata used as simplified models of nucleation and metastability in statistical physics, with connections to the Glauber dynamics of the Ising model (its standard single-spin-update Monte Carlo dynamics) and to kinetically constrained spin models (lattice spin models whose updates are blocked by local constraints, used as simplified models of the glass transition). Sites of a lattice are infected or healthy; given a finite neighbourhood $\mathcal{N}\subset\mathbb{Z}^d\setminus\{0\}$ and an integer threshold $r$, the bootstrap operator adds to an infected set $S$ every site $v$ having at least $r$ infected sites in $v+\mathcal{N}$, and iterating to infinity yields the closure $\langle S\rangle$. If the initial set is chosen $p$-randomly (each site infected independently with probability $p$) on a finite box $\Lambda$, where $[L]^2=\{1,\dots,L\}^2$ denotes the $L\times L$ box, the critical probability $p_c(\Lambda,\mathcal{N},r)$ is the value of $p$ at which the probability that every site of $\Lambda$ eventually becomes infected crosses $1/2$; the transition is sharp, so the crossing level is immaterial. A rectangle $R$ is internally filled (IF) if every site of $R$ is eventually infected using only initial infections inside $R$; this is the key large-deviation event, because a translate of an internally filled critical droplet (a rectangle of the critical dimensions) typically seeds the infection of the whole box.

The archetypal two-neighbour model ($d=2$, the four nearest neighbours, $r=2$) satisfies $p_c([L]^2)=\frac{\pi^2}{18\log L}-\frac{\Theta(1)}{(\log L)^{3/2}}$: the leading term is Holroyd's sharp threshold, and the negative second term — an entropic effect of random fluctuations of the droplet boundary around its symmetric optimal growth path — was bounded by Gravner, Holroyd and Morris and determined by Hartarsky and Morris. The anisotropic model studied in the source paper is the two-dimensional process with the $(1,2)$-neighbourhood $\mathcal{N}_{(1,2)}=\{(\pm1,0),(0,\pm1),(\pm2,0)\}$ (horizontal reach two, vertical reach one) and threshold $r=3$, introduced in the threshold-growth literature by Gravner and Griffeath. It is the canonical example of an unbalanced two-dimensional critical model: growth is much easier along one axis, the critical droplet is strongly elongated, and its optimal growth trajectory — the most probable sequence of rectangle dimensions along which a small seed grows to fill the droplet — is far from the symmetric diagonal of the two-neighbour model. The Duarte model is a neighbouring two-dimensional anisotropic model with the same leading order $\frac{(\log\log L)^2}{\log L}$ (up to constants) but a different droplet geometry; the sharpest known bounds for its critical probability, due to Bollobás, Duminil-Copin, Morris and Smith, determine only the leading order up to a constant factor, together with a sharp threshold for the metastability window.

For the anisotropic $(1,2)$-model, van Enter and Hulshof first determined the order of magnitude of $p_c$, and Duminil-Copin and van Enter then determined the first term and proved sharpness of the threshold. The source paper established the second and third terms:

$$p_c([L]^2,\mathcal{N}_{(1,2)},3)=\frac{\log\log L}{12\log L}\Big(\log\log L-4\log\log\log L+2\log\frac{9e}{2}\pm o(1)\Big),$$

making this the two-dimensional model whose critical probability is known most precisely. The determination rests on a two-term estimate of the droplet large-deviation probability. Theorem 1.3 of the paper states that, for an $x\times y$ rectangle $R$ with side lengths $1/p^2\le x\le1/p^5$ and $\frac{1}{3p}\log\frac{1}{p}\le y\le\frac{1}{p}\log\frac{1}{p}$,

$$\mathbb{P}_p(R\text{ is IF})=\exp\Big(-\frac{1}{6p}\log^2\frac{1}{p}+\Big(\frac{1}{3}\log\frac{8}{3e}\pm o(1)\Big)\frac{1}{p}\log\frac{1}{p}\Big),$$

that is, two leading terms of $\log\mathbb{P}_p(R\text{ is IF})$ with a $o(\frac{1}{p}\log\frac{1}{p})$ remainder. An iterative argument converts droplet probabilities into successive terms of $p_c$; the proof combines Holroyd's hierarchies method — a renormalised, tree-like bookkeeping of the order in which growth events occur — with a decomposition into good and bad hierarchies and a variational principle over growth trajectories. The paper also states, without full proof, a generalisation to the $(1,b)$-models (neighbourhood $\{(\pm i,0):1\le i\le b\}\cup\{(0,\pm1)\}$, threshold $b+1$):

$$p_c([L]^2,\mathcal{N}_{(1,b)},b+1)=\frac{(b-1)^2\log\log L}{4(b+1)\log L}\big(\log\log L-4\log\log\log L-C(b)\pm o(1)\big),$$

where, writing binomial coefficients with negative lower index as zero,

$$C(b)=\frac{2}{b-1}\log\Big(\frac{\binom{2b}{b}-\frac{2b-1}{b+1}\binom{2b-2}{b}-1+\binom{2b}{b-1}-\binom{2b-2}{b-3}}{(b+1)e}\Big)+2\log\Big(\frac{(b-1)^2}{4(b+1)}\Big).$$

The authors remark that Theorem 1.4 follows from the corresponding generalisation of Theorem 1.3 to the $(1,b)$-models, in which the constants $\frac{1}{6}$ and $\frac{1}{3}\log\frac{8}{3e}$ are replaced by $\frac{(b-1)^2}{2(b+1)}$ and $\frac{b-1}{b+1}\log\big(\frac{\binom{2b}{b}-\frac{2b-1}{b+1}\binom{2b-2}{b}-1+\binom{2b}{b-1}-\binom{2b-2}{b-3}}{(b+1)e}\big)$, respectively. The authors omit the proof, describing it as conceptually the same as the case $b=2$ but requiring several straightforward yet lengthy calculations.

The authors also compare their expansion with simulations and with the two-neighbour model. The third-order approximation is still farther from simulated values than the first-order one for lattices up to $L\approx10{,}000$; they suggest, without proof, that the error term in Theorem 1.3 may be as large as $O(\frac{1}{p}\log\log\frac{1}{p})$; they note that even if a further constant $C_3$ with a $\frac{C_3}{p}$ term existed, it might be impossible to determine it by fitting simulations, because the correction terms decay extremely slowly — an instance of the bootstrap percolation paradox, the systematic failure of asymptotic predictions to approximate finite-size simulations at accessible lattice sizes; and they state the heuristic belief that entropic fluctuations around the optimal trajectory — the mechanism behind the two-neighbour model's second term — do influence $p_c$ for the anisotropic model, but that their effect should be much smaller than the remaining trajectory-shape effects. The unresolved target is the next correction terms of the expansion.

Problem Statement

Determine the next-order correction terms in the metastable expansion of the two-dimensional anisotropic bootstrap percolation family, to the precision left open by Duminil-Copin, van Enter and Hulshof (2018). The problem has two linked parts serving one objective — completing the higher-order asymptotic analysis of this family.

(a) Fourth and higher-order terms (central case $b=2$). For the anisotropic $(1,2)$-model, determine the sharp order — and, when it exists, the constant — of the remainder in the critical-droplet expansion of Theorem 1.3 of the source paper (the two-term internally-filled probability estimate stated in the Background). In particular: prove or disprove that the $o(\frac{1}{p}\log\frac{1}{p})$ remainder is $O(\frac{1}{p}\log\log\frac{1}{p})$; determine whether a next-order constant $C_3$ exists, so that the droplet expansion holds with an additional $\frac{C_3}{p}$ term and error $o(\frac{1}{p})$, and if so compute $C_3$; and derive from the sharpened droplet estimates the corresponding fourth (and, if possible, further) terms in the expansion of $p_c([L]^2,\mathcal{N}_{(1,2)},3)$ beyond the three terms of Theorem 1.1. A sharp expansion necessarily accounts for the contribution of entropic fluctuations around the optimal growth trajectory — the mechanism that produces the second term in the two-neighbour model.

(b) Generalised family. Provide a complete rigorous proof of Theorem 1.4 of the source paper for every $b\ge2$: the two-term droplet estimate for the $(1,b)$-model with the constants stated there (the replacements of $\frac{1}{6}$ and $\frac{1}{3}\log\frac{8}{3e}$ given in the Background), including the binomial counts of growth configurations entering $C(b)$ and the endpoint cases (consistency with Theorem 1.1 at $b=2$; the non-growing configurations that arise for $b\ge3$), and the resulting three-term expansion of $p_c([L]^2,\mathcal{N}_{(1,b)},b+1)$.

Part (a) pushes the expansion of the canonical model one term further; part (b) establishes, in full rigour and for the whole family, the expansion currently only announced. The source discussion also records why numerical fitting cannot substitute for (a): the correction terms decay extremely slowly, so constants such as a hypothetical $C_3$ cannot be identified from simulations at accessible lattice sizes, and the recently developed locality-based exact algorithms — locality being the property, proved so far only for the Froböse and two-neighbour models, that the critical probability is determined by bounded neighbourhoods of the lattice — do not extend to this unbalanced model. The completion boundary of the problem is therefore mathematical: sharp asymptotic statements with explicit constants and controlled error terms, at least one term beyond the state of the art for the $(1,2)$-model, plus the complete proof of the $(1,b)$-family statement.

References

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