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Bootstrap percolation is a monotone version of the Glauber dynamics of the Ising model of ferromagnetism. The $r$-neighbour bootstrap process on a locally finite graph $G$ is a monotone cellular automata on the configuration space $\{0,1\}^{V(G)}$,
where the initial state is usually chosen to be the product of Bernoulli measures with density $p$.

In this talk we will consider anisotropic bootstrap models, which are three-dimensional analogues of a family of (two-dimensional) processes studied by Duminil-Copin, van Enter and Hulshof. In these models the graph $G$ has vertex set $[L]^3$, and the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $e_i$-direction for each $i \in \{1,2,3\}$,
where $a_1\le a_2\le a_3$. The main question in the area is to determine the so-called {\it critical length for percolation} $L_c(p)$, for small values of $p$.
It turns out that $L_c(p)$ is polynomial in $p$ if $r\le a_3$. On the other hand, van Enter and Fey
showed that $L_c(p)$ is doubly exponential when $r=a_1+a_2+a_3$.

In recent work, we have proved that $L_c(p)$ is singly exponential in the case $r=a_3+1$, and we will focus on this result.
To do so, we introduce a new technique called the beams process, and state an exponential decay property about cluster size distributions for two-dimensional anisotropic models that look like classical Bernoulli percolation.

Please join us to this virtual seminar at https://ucdavisdss.zoom.us/j/947840508 (room opens at 2pm)