This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise.

Schedule for

Piotr Milos, 26.04.2012


10.01.2019 Frank den Hollander (Leiden)
Exploration on dynamic networks
Search algorithms on networks are important tools for the organisation of large data sets. A key example is Google PageRank, which assigns a numerical weight to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of measuring its relative importance within the set. The weighting is achieved by exploration. The mixing time of a random walk on a random graph is the time it needs to approach its stationary distribution (also called equilibrium distribution). The characterisation of the mixing time has been the subject of intensive study. Many real-world networks are dynamic in nature. It is therefore natural to study random walks on dynamic random graphs. In this talk we focus on a random graph with prescribed degrees. We investigate what happens to the mixing time of the random walk when at each unit of time a certain fraction of the edges is randomly rewired. We identify three regimes in the limit as the graph becomes large: fast, moderate, slow dynamics. These regimes exhibit surprising behaviour.
17.01.2019 Yuri Yakubovich (St Petersburg State University)
Limit shape of minimal difference partitions and fractional quantum statistics
The class of minimal difference-q partitions MDP(q) is defined by the restriction that parts in an integer partition differ from one another by at least q > 0. Similarities between the asymptotics of random MDP(q) and quantum systems with fractional statistics were discussed by Comtet et al. by formally allowing a fractional q in [0; 1], thus interpolating between the classic Bose-Einstein (q = 0) and Fermi-Dirac (q = 1) statistics. We justify this "replica-trick" by introducing a well-defined combinatorial model, for which we rigorously derive the limit shape of MDP(q) (in the full range q in [0;1)) under the uniform measure. This will provide the first mathematical justi cation of the "fractional" limit shape.
This is joint work with Leonid Bogachev (Leeds).

24.01.2019 Jon Noel (University of Warwick)
Bootstrap percolation in high dimensions and related results
Bootstrap percolation is one of the simplest models of propagation phenomena in networks. Initially, some subset of vertices is declared "infected" and then, in each step of the process, a "healthy" vertex becomes infected if it has at least r infected neighbours. Driven by applications in physics, a main focus in bootstrap percolation has been on its behaviour in multidimensional square grids. In this talk, we will discuss some of the most important breakthroughs and open problems in this area, and mention a few of my results along the way. Joint work with Natasha Morrison.

07.02.2019 Cécile Mailler (University of Bath)
The monkey walk: a random walk with random reinforced relocations and fading memory
In this joint work with Gerónimo Uribe-Bravo, we prove and extend results from the physics literature about a random walk with random reinforced relocations. The "walker" evolves in $mathbb Z^d$ or $mathbb R^d$ according to a Markov process, except at some random jump-times, where it chooses a time uniformly at random in its past, and instantly jumps to the position it was at that random time. This walk is by definition non-Markovian, since the walker needs to remember all its past.
Under moment conditions on the inter-jump-times, and provided that the underlying Markov process verifies a distributional limit theorem, we show a distributional limit theorem for the position of the walker at large time. The proof relies on exploiting the branching structure of this random walk with random relocations; we are able to extend the model further by allowing the memory of the walker to decay with time.

Contact: Daniel
14.02.2019 Andrey Dorogovtsev (Kyiv)
Hilbert-valued Brownian self-intersection local times and geometry of compact sets
In the talk we discuss a model of behaviour of polymers in random media. The main goal is to analyse how the shape of polymer impacts its motion. Linear polymer is described by a non-smooth random curve. The self-intersection local times will be used as the geometric characteristics of the curve. To consider dynamics of polymers we have to define the self-intersections for image of the curve under a smooth random map. Such construction is based on the extension of Dynkin’s renormalization to the case of unbounded weights, and suggests to study compact sets in Hilbert space. We will focus on different characteristics of the compact sets in Hilbert space, their relation to Kolmogorov entropy, and applications to the renormalization. The talk is based on a joint work with Olga Izyumtseva.
Contact: Aleks Mijatovic (Stats), Stefan G

28.02.2019 Yu Gu (Carnegie Mellon University)
Fluctuations of the KPZ equation in d geq 2 in a weak disorder regime
We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d geq 2.
Contact: Wei

14.03.2019 Erik Slivken (Paris)
CANCELLED - rescheduled for May 9