Research interests

My research lies at the interface of probability theory and statistical physics, I am interested in the theory of bootstrap percolation and its applications. The \(r\)-neighbour bootstrap process on a locally finite graph \(G\) is a monotone cellular automaton on the configuration space \(\{0,1\}^{V(G)}\) (we call vertices in state \(1\) "infected"), evolving in discrete time in the following way: \(0\) changes to a \(1\) when it has at least \(r\) neighbours in state \(1\), and infected vertices remain  infected forever. The initial state is usually chosen to be the product of Bernoulli measures with density \(p\), and the main question is to determine the so-called  threshold for percolation: the value of \(p\) above which the entire vertex set is likely to be infected by the end of the process.
I have considered a family of \(d\)-dimensional models known as anisotropic bootstrap percolation. In  these models the graph \(G\) has vertex set \([L]^d\), and the neighbourhood of each vertex consists of the \(a_i\) nearest neighbours in the \(e_i\)-direction for each \(i\in [d]\). I  have so far been able to determine the threshold for percolation in the case \(d=3\) and \(r = \max\{a_i\} + 1\).
In a seminal work, Fontes, Schonmann and Sidoravicius, used techniques from the study of bootstrap percolation to show the  existence of a phase transition for the zero-temperature Glauber dynamics of the Ising model on \(\mathbb Z^d\). I have proved an analogous  theorem for the so-called \(\mathcal U\)-voter dynamics, for various families \(\mathcal U\).

 

Preprints

  • Anisotropic bootstrap percolation in three dimensions. Submitted[arXiv link].
  • Fixation for two-dimensional \(\,\mathcal U\)-voter dynamics. In preparation. [pdf]