Students joining SAMBa in 2015

###### Andrea Lelli

Andrea graduated with an MSc degree in Mathematics from the University of Rome "Roma Tre", in 2015.

Andrea graduated with an MSc degree in Mathematics from the University of Rome “Roma Tre”, in 2015. He specialized in probability and, more precisely, in the theory of Markov chain mixing. His master level thesis was based on studying the problem of the Cutoff phenomena for the Ising model on finite graphs at high temperature. In his spare time, Andrea enjoys watching movies, acting in theatre or playing his little ukulele.

**Research project title:**

Mixing times and general behavior of random walks on changing environments

**Supervisor(s):**

Alexandre Stauffer

**Project description:**

Random walks in random environments have become a classical model for random motion in random media, and this model has been the source of many mathematical investigations over the years. More recently, people started to look at random walks in an environment which changes at the same time that the particle is moving. It is believed that when the environment is ‘well behaved’ (e.g. uniformly elliptic) and changes quickly enough, the random walk will behave in a way that is similar to a random walk on the underlying (non-changing) graph. This has been quantified, especially in the case of the d-dimensional infinite lattice, by the derivation of a law of large numbers and central limit theorems under some conditions related to the mixing time of the environment. Andrea was interested in understanding the effect of a slowly changing environment on the behaviour of simple random walks, e.g. the impact of the environment on the recurrence/transience property of the random walk and the mixing time of the random walk inside a finite, but changing graph.