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We revisit the Bootstrap Percolation model, leveraging recent mathematical advances linking it with its local counterpart. This new perspective resolves, for the first time, historic discrepancies between Monte Carlo simulations and theoretical results: previously, those predictions disagreed even in the first-order asymptotics of the model. In contrast, our framework achieves excellent agreement between numerics and theory, which now match up to the third-order expansion, as the infection probability approaches zero. Our algorithm allows us to generate novel predictions for the model., Comment: 6 pages, 2 figures, 3 page supplemental material

In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i

Metastability thresholds lie at the heart of bootstrap percolation theory. Yet proving precise lower bounds is notoriously hard. We show that for two of the most classical models, two-neighbour and Frob\"ose, upper bounds are sharp to essentially arbitrary precision, by linking them to their local counterparts. In Frob\"ose bootstrap percolation, iteratively, any vertex of the square lattice that is the only healthy vertex of a $1\times1$ square becomes infected and infections never heal. We prove that if vertices are initially infected independently with probability $p\to0$, then with high probability the origin becomes infected after \[\exp\left(\frac{\pi^2}{6p}-\frac{\pi\sqrt{2+\sqrt2}}{\sqrt p}+\frac{O(\log^2(1/p))}{\sqrt[3]p}\right)\] time steps. We achieve this by proposing a new paradigmatic view on bootstrap percolation based on locality. Namely, we show that studying the Frob\"ose model is equivalent in an extremely strong sense to studying its local version. As a result, we completely bypass Holroyd's classical but technical hierarchy method, yielding the first term above and systematically used throughout bootstrap percolation for the last two decades. Instead, the proof features novel links to large deviation theory, eigenvalue perturbations and others. We also use the locality viewpoint to resolve the so-called bootstrap percolation paradox. Indeed, we propose and implement an exact (deterministic) algorithm which exponentially outperforms previous Monte Carlo approaches. This allows us to clearly showcase and quantify the slow convergence we prove rigorously. The same approach applies, with more extensive computations, to the two-neighbour model, in which vertices are infected when they have at least two infected neighbours and do not recover. We expect it to be applicable to a wider range of models and correspondingly conclude with a number of open problems., Comment: 61 pages, 8 figures, 3 tables, supplementary material included in the source and available at https://github.com/augustoteixeira/bootstrap

Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on $\mathbb R^d$ with intensity $\lambda>0$ is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate $\alpha>0$, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of $\lambda$ and $\alpha$, the infection almost surely dies out., Comment: 8 pages

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes $[n]^d$. These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on $[n]^d$ features order $n^{d-2}$ alternating cycles of length at most $4d-2$. We also prove that the dynamics of dimer configurations on the unit hypercube of dimension $d$ is ergodic when switching alternating cycles of length at most $4d-4$. Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and R\'emila, which also uses 8-cycles. None of our proofs make reference to height functions., Comment: 15 pages, 4 figures, minor adjustments of the presentation

In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $\tau$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to0$ it holds that \[\tau\le \exp\left(\frac{\pi^2}{6p}-\frac{c\log(1/p)}{\sqrt p}\right)\] for some positive constant $c$, while the classical model is known to lack the logarithmic factor., Comment: 6 pages, 1 figure, minor adjustements of the presentation

We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This extends previous instances proved for several specific rules. The paper supersedes a draft by Alexander Holroyd and the first author from 2012. While it served a role in the subsequent development of bootstrap percolation universality, we have chosen to adopt a more contemporary viewpoint in its present form., Comment: 37 pages, 6 figures, improved presentation, added section 6

We study the full class of kinetically constrained models in arbitrary dimension and out of equilibrium, in the regime where the density $q$ of facilitating sites in the equilibrium measure (but not necessarily in the initial measure) is close to $1$. For these models, we establish exponential convergence to equilibrium in infinite volume and linear time precutoff in finite volume with appropriate boundary condition. Our results are the first out-of-equilibrium results that hold for any model in the so-called critical class, which is covered in its entirety by our treatment, including e.g. the Fredrickson-Andersen 2-spin facilitated model. In addition, they generalise, unify and sometimes simplify several previous works in the field. As byproduct, we recover and generalise exponential tails for the connected component of the origin in the upper invariant trajectory of perturbed cellular automata and in the set of eventually infected sites in subcritical bootstrap percolation models. Our approach goes through the study of cooperative contact processes, last passage percolation, Toom contours, as well as a very convenient coupling between contact processes and kinetically constrained models., Comment: 40 pages, 1 figure, improved presentation

We show that for every $r\ge 3$, the maximal running time of the $K^{r}_{r+1}$-bootstrap percolation in the complete $r$-uniform hypergraph on $n$ vertices $K_n^r$ is $\Theta(n^r)$. This answers a recent question of Noel and Ranganathan in the affirmative, and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form $r^{-r} \mathrm{e}^{O(r)}$ as $r\to\infty$., Comment: 10 pages, 2 figures, updated creative commons license

We analyse the class of convex functionals $\mathcal E$ over $\mathrm{L}^2(X,m)$ for a measure space $(X,m)$ introduced by Cipriani and Grillo and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if $\mathcal E(\phi \circ f) \leq \mathcal E(f)$ for all $f \in \mathrm{L}^2(X,m)$, and all 1-Lipschitz functions $\phi: \mathbb R \to \mathbb R$ with $\phi(0)=0$. We prove that normal contraction holds if and only if $\mathcal E$ is symmetric in the sense $\mathcal E(-f) = \mathcal E(f),$ for all $f \in \mathrm{L}^2(X,m).$ An auxiliary result, which may be of independent interest, states that it suffices to establish the normal contraction property only for a simple two-parameter family of functions $\phi$., Comment: 21 pages, 8 figures, improved presentation

The present expository article overviews recent mathematical advances on the Fredrickson--Andersen kinetically constrained spin model in two dimensions. It was introduced in physics as a toy model for recovering the glassy phenomenology in supercooled liquids close to the glass transition via dynamic constraints as opposed to static interactions., Comment: 11 pages, 1 figure, journ\'ees MAS 2020. arXiv admin note: text overlap with arXiv:2012.02557

In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom's North-East-Center majority rule cellular automaton., Comment: 16 pages, 3 figures

We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a classical result of Toom on the stability of cellular automata w.r.t. noise and, inversely, how to deduce new results in bootstrap percolation universality from his work., Comment: 22 pages, improved presentation

The bisection method for kinetically constrained models (KCM) of Cancrini, Martinelli, Roberto and Toninelli is a vital technique applied also beyond KCM. In this note we present a new way of performing it, based on a novel two-block dynamics with a probabilistic proof instead of the original spectral one. We illustrate the method by very directly proving an upper bound on the relaxation time of KCM like the one for the East model in a strikingly general setting. Namely, we treat KCM on finite or infinite one-dimensional volumes, with any boundary condition, conditioned on any of the irreducible components of the state space, with arbitrary site-dependent state spaces and, most importantly, arbitrary inhomogeneous rules., Comment: 9 pages, additional details, improved presentation, extended section 2

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions. They are tightly linked to the monotone cellular automata called bootstrap percolation. Among the three classes of such models, the critical ones are the most studied. Together with the companion paper by Mar\^ech\'e and the author, our work determines the logarithm of the infection time up to a constant factor for all critical KCM. This was previously known only up to logarithmic corrections. We establish that on this level of precision critical KCM have to be classified into seven categories. This refines the two classes present in bootstrap percolation and the two in previous rougher results. In the present work we establish the upper bounds for the novel five categories and thus complete the universality program for equilibrium critical KCM. Our main innovations are the identification of the dominant relaxation mechanisms and a more sophisticated and robust version of techniques recently developed for the study of the Fredrickson-Andersen 2-spin facilitated model., Comment: 100 pages, including 3 online-only appendices, 10 figures; comprehensive summary of mechanisms in section 2; changes: none (updated creative commons license)

We consider a generalised oriented site percolation model (GOSP) on $\mathbb Z^d$ with arbitrary neighbourhood. The key additional difficulties as compared to standard oriented percolation (OP) are the lack of symmetry and, in two dimensions, of planarity. We establish that, despite these deficiencies, in the supercritical regime GOSP behaves qualitatively like OP., Comment: 33 pages, 6 figures, improved presentation in section A.1

The Fredrickson-Andersen 2-spin facilitated model on $\mathbb{Z}^d$ (FA-2f) is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring dynamical facilitation, an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbours are empty. Although the process is reversible w.r.t. a product Bernoulli measure, it is not attractive and features degenerate jump rates and anomalous divergence of characteristic time scales as the density $q$ of empty sites tends to $0$. A natural random variable encoding the above features is $\tau_0$, the first time at which the origin becomes empty for the stationary process. Our main result is the sharp threshold \[\tau_0=\exp\Big(\frac{d\cdot\lambda(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad \text{w.h.p.}\] with $\lambda(d,2)$ the sharp threshold constant for 2-neighbour bootstrap percolation on $\mathbb{Z}^d$, the monotone deterministic automaton counterpart of FA-2f. This is the first sharp result for a critical KCM and it compares with Holroyd's 2003 result on bootstrap percolation and its subsequent improvements. It also settles various controversies accumulated in the physics literature over the last four decades. Furthermore, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM., Comment: 41 pages, 5 figures, minor changes in section 2

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions tightly linked to the monotone cellular automata called bootstrap percolation. There are three classes of such models, the most studied being the critical one. In a recent series of works it was shown that the KCM counterparts of critical bootstrap percolation models with the same properties split into two classes with different behaviour. Together with the companion paper by the first author, our work determines the logarithm of the infection time up to a constant factor for all critical KCM, which were previously known only up to logarithmic corrections. This improves all previous results except for the Duarte-KCM, for which we give a new proof of the best result known. We establish that on this level of precision critical KCM have to be classified into seven categories instead of the two in bootstrap percolation. In the present work we establish lower bounds for critical KCM in a unified way, also recovering the universality result of Toninelli and the authors and the Duarte model result of Martinelli, Toninelli and the second author., Comment: 56 pages, 3 figures; minor changes

We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $\kappa$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $\kappa -1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most nontrivial values of $\kappa$. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically $1/(2d)$. For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest., Comment: 23 pages, 2 figures

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-$1$f) KCM on the discrete $d$-dimensional torus. In view of applications to the more complex FA-$j$f KCM, $j>1$, we also extend part of the analysis to an analogous process with a more general product state space., Comment: 19 pages, minor changes

In this paper we consider kinetically constrained models (KCM) on $\mathbb Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to the so-called "critical class" our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the present authors proved that if $\mathcal U$ has an infinite number of "stable directions", then on a doubly logarithmic scale the above divergence is twice the one in the corresponding $\mathcal U$-bootstrap percolation. Here we prove instead that, contrary to previous conjectures, in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical $\mathcal U$. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions., Comment: 36 pages, 11 figures, expanded and revised presentation, added roadmaps for sections 4 and 5, added and improved figures

Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb{Z}^d$ with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial state known as $\mathcal{U}$-bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics. In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes. Here we study in full generality the class termed `critical'. Together with the companion paper by Martinelli and two of the authors we establish the universality classes of critical KCM and determine within each class the critical exponent of the infection time as well as of the spectral gap. In this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart. This is due to the occurrence of `energy barriers', which determine the dominant behaviour for these KCM but which do not matter for the monotone bootstrap dynamics. Our result confirms the conjecture of Martinelli, Morris and the last author, who proved a matching upper bound., Comment: 33 pages, 6 figures, expanded section 5.1, minor improvements of the presentation

We study the evolution of the maximum energy $E_\max(t)$ reached between time $0$ and time $t$ in the dynamics of simple models with glassy energy landscapes, in instant quenches from infinite temperature to a target temperature $T$. Through a detailed description of the activated dynamics, we are able to describe the evolution of $E_\max(t)$ from short times, through the aging regime, until after equilibrium is reached, thus providing a detailed description of the long-time dynamics. Finally, we compare our findings with numerical simulations of the $p$-spin glass and show how the maximum energy record can be used to identify the threshold energy in this model., Comment: 22 pages, 5 figures; revised presentation

Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three rough universality classes, the most studied being the `critical' one. For this class the scaling of the quantity of greatest interest -- the critical probability -- was determined by Bollob\'as, Duminil-Copin, Morris and Smith in terms of a simply defined combinatorial quantity called `difficulty', so the subject seemed closed up to finding sharper results. However, the computation of the difficulty, was never considered. In this paper we provide the first algorithm to determine this quantity, which is, surprisingly, not as easy as the definition leads to thinking. The proof also provides some explicit upper bounds, which are of use for bootstrap percolation. On the other hand, we also prove the negative result that computing the difficulty of a critical model is NP-hard. This two-dimensional picture contrasts with an upcoming result of Balister, Bollob\'as, Morris and Smith on uncomputability in higher dimensions. The proof of NP-hardness is achieved by a technical reduction to the Set Cover problem., Comment: 21 pages, 2 figures, changes: improved presentation, detailed proofs, new appendix

Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities and successfully apply this link to prove new and old results for concrete models such as DTBP and Spiral as well as a general non-trivial upper bound. Our approach establishes and exploits a tight connection between subcritical bootstrap percolation and a suitable generalisation of classical oriented percolation, which will undoubtedly be the source of more results and could provide an entry point for general percolationists to bootstrap percolation. Furthermore, we prove that above a certain critical probability there is exponential decay of the probability of a one-arm event, while below it the event has positive probability and the expected infection time is infinite. We also identify this as the transition of the spectral gap and mean infection time of the corresponding kinetically constrained model. Finally, we essentially characterise the noise sensitivity properties at fixed density for the two natural one-arm events. In doing so we answer fully or partially most of the open questions asked by Balister, Bollob\'as, Przykucki and Smith (2016)---namely we are concerned with their Questions 11, 12, 13, 14 and 17., Comment: 43 pages, 5 figures; revised presentation, minor corrections

In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid-glass transition, the most extensively-studied case is the two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15 years ago, in a major breakthrough, Holroyd determined the sharp threshold for percolation in this model, and his bounds were subsequently sharpened further by Gravner and Holroyd, and by Gravner, Holroyd and Morris. In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by proving that the critical probability $p_c\big( [n]^2,2 \big)$ for percolation in the two-neighbour model on $[n]^2$ satisfies \[p_c\big( [n]^2,2 \big) = \frac{\pi^2}{18\log n} - \frac{\Theta(1)}{(\log n)^{3/2}}\,.\] The proof of this result requires a very precise understanding of the typical growth of a critical droplet, and involves a number of technical innovations. We expect these to have other applications, for example, to the study of more general two-dimensional cellular automata, and to the $r$-neighbour process in higher dimensions., Comment: 53 pages, 6 figures, 1 appendix

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(5)},\mathcal{H})$ in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank. In our first result, we construct a graph $G$ (in fact $G=K_6\setminus K_4$) and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of this result we deduce our second result in which we construct a $4$-uniform hypergraph $G'$ and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves the result in the paper above. An equivalent formulation of our first result is that the game $\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for interest on the board $K_\omega\sqcup K_\omega$ is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph $H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$ if $\mathcal{R}(K_\omega,H)$ is a first player win, then $\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win., Comment: 17 pages, 48 figures; improved presentation, particularly in section 3

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ are infected, round-by-round, if they have $r$ neighbours already infected. Once infected, they remain infected. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki. Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem., Comment: 14 pages, 1 figure, submitted