Paul Smith

CNPq Postdoctoral Research Fellow

Instituto de Matemática Pura e Aplicada
Rio de Janeiro, Brasil

Research + Background

I am interested in problems in all areas of probabilistic combinatorics, random graphs, discrete probability and percolation, especially those with a connection to statistical physics.

Before returning to IMPA in 2017, I was a Postdoctoral Research Fellow at Tel Aviv University from 2016 to 2017, a Fellow and College Lecturer at Murray Edwards College, Cambridge from 2014 to 2016, and a CNPq Postdoctoral Research Fellow (for the first time) at IMPA from 2012 to 2014. I did my PhD under the supervision of Béla Bollobás in Cambridge, at Trinity College, where I was also an undergraduate.

Papers

  • The sharp threshold for the Duarte model
    • with Béla Bollobás, Hugo Duminil-Copin, and Robert Morris
    • To appear, Ann. Probab.   |  arXiv
  • Nucleation and growth in two dimensions
    • with Béla Bollobás, Simon Griffiths, Robert Morris, and Leonardo Rolla
    • Preprint   |  arXiv
  • The threshold for jigsaw percolation on random graphs
    • with Béla Bollobás, Oliver Riordan, and Erik Slivken
    • Electron. J. Combin. 24 (2017), no. 2, 14pp.   |   arXiv   |   journal
  • Universality of two-dimensional critical cellular automata
    • with Béla Bollobás, Hugo Duminil-Copin, and Robert Morris
    • To appear, Proc. Lond. Math. Soc.   |  arXiv
  • Subcritical U-bootstrap percolation models have non-trivial phase transitions
    • with Paul Balister, Béla Bollobás, and Michał Przykucki
    • Trans. Amer. Math. Soc. 368 (2016), 7385–7411   |  arXiv   |   journal
  • The sharp threshold for maximum-size sum-free subsets in even-order abelian groups
    • with Neal Bushaw, Maurício Collares Neto, and Robert Morris
    • Combin. Probab. Comput. 24 (2015), no. 4, 609–640   |   arXiv   |   journal
  • The time of bootstrap percolation in two dimensions
    • with Paul Balister, and Béla Bollobás
    • Probab. Theory Related Fields 166 (2016), no. 1, 321–364   |  arXiv   |   journal
  • The time of bootstrap percolation with dense initial sets for all thresholds
    • with Béla Bollobás, and Andrew Uzzell
    • Random Structures Algorithms 47 (2015), no. 1, 1–29   |   arXiv   |  journal
  • The time of bootstrap percolation with dense initial sets
    • with Béla Bollobás, Cecilia Holmgren, and Andrew Uzzell
    • Ann. Probab. 42 (2014), no. 4, 1337–1373   |   arXiv   |   journal
  • Monotone cellular automata in a random environment
    • with Béla Bollobás, and Andrew Uzzell
    • Combin. Probab. Comput. 24 (2015), no. 4, 687–722   |   arXiv   |  journal

Simulations

Here are some simulations of four two-dimensional monotone cellular automata in a random environment: the classical two-neighbour bootstrap percolation model and the Duarte model (which are both critical, in the language of this paper), the subcritical directed triangular bootstrap percolation (DTBP) model (see this paper on subcritical models), and a toy supercritical model. The precise defintion of each model is given on the simulations homepage.

The simulator runs the models on 300 x 300 tori and may be modified by changing the values of the variables in the query string: density is what you think it is; type refers to the model; rate is the number of sites that are updated in each new frame. The simulator was written in Javascript / jQuery by Alex Holyoake, to whom I am greatly indebted.

Talk

In July 2016, I gave a talk at the Isaac Newton Institute in Cambridge, entitled ‘Towards universality in bootstrap percolation’. That talk may be viewed online.

Nespresso Index

I have put together a comparison of the prices of Nespresso capsules around the world.