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Register a new userBootstrap percolation in three dimensions
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By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices infected at time 0, new vertices are subsequently infected, at each time step, if they have at least $rinmathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the $d$-dimensional grid $[n]^d$: with $2leq rleq d$ fixed, it was proved by Cerf and Cirillo (for $d=r=3$), and by Cerf and Manzo (in general), that [p_c([n]^d,r)=Thetabiggl(frac{1}{log_{(r-1)}n}biggr)^{d-r+1},] where $log_{(r)}$ is an $r$-times iterated logarithm. However, the exact threshold function is only known in the case $d=r=2$, where it was shown by Holroyd to be $(1+o(1))frac{pi^2}{18log n}$. In this paper we shall determine the exact threshold in the crucial case $d=r=3$, and lay the groundwork for solving the problem for all fixed $d$ and $r$.
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In the polluted bootstrap percolation model, vertices of the cubic lattice $mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,qto 0$. We show that this density converges to $1$ if $q ll p^3(log p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists $C$ such that the final density converges to $0$ if $q > Cp^3$. For the standard model, we establish convergence to $0$ under the stronger condition $q>Cp^2$.
We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a non-trivial p_c. On the augmented triangular lattice we find, by extensive numerical simulation, that the directed rigidity percolation transition belongs to the same universality class as directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e. the spreading of rigidity from finite clusters close to a non-rigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.
We study the mixed system of correlation functions involving a scalar field charged under a global $U(1)$ symmetry and the associated conserved spin-1 current $J_mu$. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. We then specialize to the $O(2)$ model and extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the $O(2)$ model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors.
Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph~$G$ begin in one of two states, dormant or active. Given a fixed integer $r$, a dormant vertex becomes active if at any stage it has at least $r$ active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices $A$, we say that $G$ $r$-percolates (from $A$) if every vertex in $G$ becomes active after some number of steps. Let $m(G,r)$ denote the minimum size of a set $A$ such that $G$ $r$-percolates from $A$. Bootstrap percolation has been studied in a number of settings, and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree-based density conditions that ensure $m(G,2)=2$. In particular, we give an Ore-type degree sum result that states that if a graph $G$ satisfies $sigma_2(G)ge n-2$, then either $m(G,2)=2$ or $G$ is in one of a small number of classes of exceptional graphs. We also give a Chv{a}tal-type degree condition: If $G$ is a graph with degree sequence $d_1le d_2ledotsle d_n$ such that $d_i geq i+1$ or $d_{n-i} geq n-i-1$ for all $1 leq i < frac{n}{2}$, then $m(G,2)=2$ or $G$ falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D. Reichman, Contagious sets in dense graphs, to appear in European J. Combin.]
We study incidence problems involving points and curves in $R^3$. The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz, requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies, by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for point-curve incidence problems in $R^3$. Incidences of this kind have been considered in several previous studies, starting with Guth and Katzs work on points and lines. Our results, which are based on the work of Guth and Zahl concerning surfaces that are doubly ruled by curves, provide a grand generalization of most of the previous results. We reconstruct the bound for points and lines, and improve, in certain significant ways, recent bounds involving points and circles (in Sharir, Sheffer and Zahl), and points and arbitrary constant-degree algebraic curves (in Sharir, Sheffer and Solomon). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge. As an application of our point-curve incidence bound, we show that the number of triangles spanned by a set of $n$ points in $R^3$ and similar to a given triangle is $O(n^{15/7})$, which improves the bound of Agarwal et al. Our results are also related to a study by Guth et al.~(work in progress), and have been recently applied in Sharir, Solomon and Zlydenko to related incidence problems in three dimensions.
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