
Professor Marton Balazs
MSc, MSc, Phd
Expertise
A probabilist working mostly with interacting particle systems and other stochastic processes.
Current positions
Professor of Probability
School of Mathematics
Contact
Press and media
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Research interests
Please see my personal webpage for information on my research.
Available PhD projects
Various phenomena in last passage percolation
Place i.i.d. random weights on the vertices of the planar lattice Z^2, and consider two points in that lattice. The first passage percolation (FPP) problem asks about the paths on Z^2 between these two points that collect the least total weight along the way. Questions include existence, uniqueness, and geometric properties (e.g., fluctuations from the straight line between the two points) of such paths, as well as asymptotics (Law of Large Numbers, fluctuations) of the total weight collected by them. While this model is very natural, even some of the most basic questions seem very hard.
The last passage percolation problem asks the same questions, but for the paths collecting the largest total weight along the way. To make sense of this, one restricts the relative position of the two points to North-East, as well as the possible steps of the paths to either North or East on the lattice. Suddenly more structure is available and many more questions have been answered than in FPP. There are still things to explore, this is what this topic will be about. We'll use probabilistic arguments, admiration for those will be useful.
Fluctuations in interacting particle systems
In this field particles are placed on the sites of the integer line Z, and a stochastic dynamics is run on the resulting configurations. The main feature, which makes these models both interesting and difficult at the same time, is that the particles influence each other, hence the word "interacting" in the title. Under rather general assumptions the models exhibit non-conventional scaling properties: rather than square-root scaling and Normal distributional limits, one often finds one-third power of scaling and other limit distributions. This has been proved for a handful of models and not yet proved for many others in the area. This project will concentrate on using probabilistic arguments to prove sharper and/or more general results than available on such exotic scalings.
Projects and supervisions
Research projects
Particle systems, growth models and their probabilistic structures
Principal Investigator
Managing organisational unit
School of MathematicsDates
05/12/2022 to 04/12/2025
Stochastic interacting systems: connections, fluctuations and applications
Principal Investigator
Managing organisational unit
School of MathematicsDates
01/06/2018 to 20/05/2022
Thesis supervisions
Publications
Recent publications
11/02/2022Hydrodynamic limit of the zero range process on a randomly oriented graph
Electronic Journal of Probability
Interacting Particle Systems and Jacobi style identities
Research in the Mathematical Sciences
Local stationarity in exponential last-passage percolation
Probability Theory and Related Fields
Non-existence of bi-infinite geodesics in the exponential corner growth model
Forum of Mathematics, Sigma
Large deviations and wandering exponent for random walk in a dynamic beta environment
Annals of Probability