We report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation.
Initially introduced as toy model for quantum diffusion, the non-linear supersymmetric hyperbolic sigma model has been attracting much attention in recent years due to its connection to history dependent stochastic processes. In this talk I will present a version of the model with hierarchical interactions. The internal symmetries of the model allow to perform some block-spin renormalization steps exactly. The resulting effective action has renormalized coefficients but no additional interaction terms. I will show the corresponding derivation and some applications.
Geometric methods proved to be useful in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric objects such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymptototic behavior of them shows surprising similarity. Random walks is an essential tool in studying large-scale geometry of groups. On the other hand it is an interesting object for probabilists since many properties of general stochastic processes are manifested here in a rather simple form. In my talk, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.
An LMS online lecture course in Markov chains.
We will start the course by presenting various results using the semimartingale approach for Markov chains. These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent. We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.
1. Irrational rotations on torus;
2. Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
3. Furstenberg's ×2,×3 theorem;
4. Results and problems on digit expansions of integers;
5. Furstenberg's theorem on 2-dimensional torus (if time permits).
Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.
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