Fractional Stochastic Calculus via Stochastic Sewing

**Thema der Dissertation:**Fractional Stochastic Calculus via Stochastic Sewing

On random interlacements

**Thema der Disputation:**On random interlacements

In the first part of the talk, a pedagogical overview of random interlacements will be provided.

Especially, we discuss the phase transition of the model with respect to the parameter $u$; that is, for small $u$ the complement of random interlacements possesses a unique unbounded component, while all components are finite for large $u$. In the second part of the talk, we briefly review another innovative work (Invent. Math, 2023) by Sznitman on the asymptotic probability regarding regions disconnected by random interlacements.

**Abstract:**In the seminal work (Ann. of Math, 2010), Sznitman introduced a model of random interlacements that consists of doubly infinite trajectories on $\mathbb{Z}^d$ ($d ≥ 3$), where trajectories are essentially sampled from those of two-sided simple random walk. Random interlacements arise as the limiting distribution of the trace of simple random walk on the torus of size $N$ run by time $u \times N^d$, with some positive parameter $u$. Alternatively, they can be defined through certain Poisson point process on the space of trajectories on $\mathbb{Z}^d$, with the parameter $u$ measuring how many trajectories come into the picture.In the first part of the talk, a pedagogical overview of random interlacements will be provided.

Especially, we discuss the phase transition of the model with respect to the parameter $u$; that is, for small $u$ the complement of random interlacements possesses a unique unbounded component, while all components are finite for large $u$. In the second part of the talk, we briefly review another innovative work (Invent. Math, 2023) by Sznitman on the asymptotic probability regarding regions disconnected by random interlacements.

### Time & Location

Sep 12, 2023 | 10:00 AM

Seminarraum 031

(Fachbereich Mathematik und Informatik, Arnimallee 7, 14195 Berlin)