# 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 $$\mathcal U$$.

### Publications and preprints

• Anisotropic bootstrap percolation in three dimensions. Annals of ProbabilityVolume 48, Number 5 (2020), 2591-2614.
• Fixation for two-dimensional $$\,\mathcal U$$-Ising and $$\,\mathcal U$$-Voter dynamics. Submitted. [arXiv link]
• On heterogeneous bootstrap percolation. In preparation.