My current research in Cambridge focuses on the analysis of a one-parameter family of random growth models on the complex plane, so called Hastings-Levitov models. These consist of growing fractal-type clusters on the plane, and are believed to include Diffusion Limited Aggregation (DLA), the Eden model and Dielectric Breakdown models. For one particular choice of the parameter, James Norris and Amanda Turner showed that the limiting shape of large clusters is a disc: in [3] I analyse fluctuations around this asymptotic behaviour, showing that they are described by a distribution-valued stochastic process which, as the clusters are left to grow indefinitely, converges to the restriction of a whole plane Gaussian Free Field to the unit circle {|z|=1}.

Here are some very nice videos of Hastings-Levitov growth made by Henry Jackson: HL(0), HL(1), HL(2) (recommended to watch in HD!).

On a more applied note, I am also working with Alessandra Faggionato on the mathematical modelling of a class of proteins, so called molecular motors, performing fundamental tasks in living cells. In recent years, the problem of obtaining an effective description of the motion of these proteins has received much attention in the biophysical literature: we provide in [1]-[2] a rigorous approach, with a particular focus on large fluctuations and Gallavotti–Cohen type symmetries. 


[1] Faggionato A., Silvestri V.: Random walks on quasi one dimensional lattices: large deviations and fluctuation theorems. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques (2017) Vol.53, No. 1, 46-78.

[2] Faggionato A., Silvestri V.: Discrete kinetic models for molecular motors: asymptotic velocity and gaussian fluctuations. Journal of Statistical Physics 157.6 (2014): 1062-1096.

[3] Silvestri V.: Fluctuation results for Hastings-Levitov planar growth. Probability Theory and Related Fields (2015): 1-44.

[4] Faggionato A., Silvestri V.: Fluctuation theorems for discrete kinetic models of molecular motors. Journal of Statistical Mechanics: Theory and Experiments (2017): 043206. 


[5] Levine L., Silvestri V.: How long does it take for Internal DLA to forget its initial profile? Submitted.