A/Prof Andrea Collevecchio

Research Overview

In principle I am interested in any topic involving randomness: large deviations, mixing time for Markov chains, game theory, long memory processes, preferential attachment models.

* Reinforced Random Walks I study long-memory processes, e.g. reinforced random walks, where the sites that are frequently visited in the past are more likely to be visited in the future. These processes are related to ant-based algorithms where the paths related to “better outcomes” are given larger weights for future iterations. The ant-based algorithms are very useful in optimization. A recent result that I obtained together with Tuan-Minh Nguyen and Stas Volkov (https://arxiv.org/abs/2004.05927) is that Vertex-Reinforced random walk on Z with strong reinforcement localises on exactly 3 sites, and only one of them has an unbounded local time.

* Game Theory I recently studied games with a large number of players and random payoffs. Up to this stage I focused on independent payoffs, and had very detailed results on the geometry of Nash Equilibria, thanks to a connection we made between these games and percolation on the hypercube. (https://arxiv.org/abs/1905.10758)

Selected Publications

[1] Collevecchio, Andrea ; Kious, Daniel ; Sidoravicius, Vladas. "The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees". In: Communications on Pure and Applied Mathematics. 2020 ; Vol. 73, No. 1. pp. 210-236. https://doi.org/10.1002/cpa.21860

[2] Collevecchio, Andrea ; Hamza, Kais ; Tournier, Laurent. "A deterministic walk on the randomly oriented Manhattan lattice". In: Electronic Journal of Probability. 2019 ; Vol. 24. https://doi.org/10.1214/19-EJP385 

[3] Collevecchio, Andrea ; Elçi, Eren Metin ; Garoni, Timothy M. ; Weigel, Martin. "On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model". In: Journal of Statistical Physics. 2018 ; Vol. 170, No. 1. pp. 22-61. https://doi.org/10.1007/s10955-017-1912-x

[4] Collevecchio, Andrea ; Konig, Wolfgang ; Morters, Peter ; Sidorova, Nadia. "Phase transitions for dilute particle systems with Lennard-Jones potential". In: Communications in Mathematical Physics. 2010 ; Vol. 299, No. 3. pp. 603 - 630. https://doi.org/10.1007/s00220-010-1097-5

[5] Collevecchio, Andrea. "On the transience of processes defined on Galton-Watson trees". In: Annals of Probability. 2006 ; Vol. 34, No. 3. pp. 870-878. https://doi.org/10.1214/009117905000000837