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# Colloquium in probability

Organisers: Nina Gantert (TUM), Noam Berger (TUM), Markus Heydenreich (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Konstantinos Panagiotou (LMU), Sabine Jansen (LMU),

TBA
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## Previous talks

### 15.03.2023 16:30 Nico Baierlein, LMU (MSc presentation): Sharp phase transition in the random connection model

We consider the random connection model with bounded edges which is generated by a Poisson point process with density $\lambda$ in $\mathbb{R}^d$. We prove that this model undergoes a sharp phase transition, i.e. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to ($\lambda-\lambda_c)$, $\lambda_c$ being the critical density. This proof uses newly developed methods by Last, Peccati and Yogeshwaran in their recent work, in particular a continuous version of the OSSS inequality for Poisson functionals, relying on stopping sets and continuous-time decision trees. This approach simplifies an earlier result of Faggionato and Mimun, who proved sharp phase transition in the random connection model via the discrete OSSS inequality.
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### 13.02.2023 15:30 David Zettler: Miller-Abrahams random resistor network

Based on a collection of works by Alessandra Faggionato, I will give a short introduction to Mott's Law and its rigorous derivation. It states that the conductivity in amorphous materials scales like $\exp( -c\beta^{1/4})$ for low temperatures, where $\beta$ denotes the inverse temperature. Using the Miller-Abrahams resistor network, A. Faggionato developed two approaches to rigorously prove such a limiting behavior. One via scaling limit of the conductivity of random resistor networks on simple point processes, and one via critical conductance of the Miller-Abrahams resistor network. A recent work of her presented at Paris CIRS connects these two approaches, showing that they both lead to the same sub-exponential decay of conductivity. Although, for a complete proof, a lower bound on LR-crossings in the supercritical regime for energy marks of both signs is still missing, and I will comment on this issue.
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### 13.02.2023 16:30 Sam Olesker-Taylor (University of Warwick): Metastability for Loss Networks

We consider a fully-connected loss network with dynamic alternative routing, each link of capacity K. Calls arrive to each link {i, j} at rate λ independently and depart at rate 1. If the link is full upon arrival, a third node k is chosen uniform and the call is routed via k: it uses a unit of capacity on both {i, k} and {k, j} if both have spare capacity; otherwise, the call is lost. This is a model for telephone networks, implemented by BT in the 1990s. We analyse the asymptotics of the mixing time of this process, depending on the traffic intensity α := λ/K. In particular, we determine a phase transition at an explicit threshold α*: there is fast mixing if α < α* or α > 1, but metastability if α* < α < 1. We also discuss a fixed for metastability—ie, an adjustment to the model which removes the slow-mixing phase. Again, this was implemented by BT in the UK telephone network.
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### 06.02.2023 16:30 Markus Lobenwein (LMU) "MSc Presentation": Diffusionen auf Mannigfaltigkeiten für die Steinsche Methode der austauschbaren Paare

Ziel des Vortrags ist es, mithilfe von Diffusionen auf Riemann Mannig- faltigkeiten eine Familie von austauschbaren Paaren im R^d für die Steinsche Meth- ode der austauschbaren Paare zu konstruieren und daraus eine Abschätzung zu gewinnen. Um die nötigen Schritte zu erklären, werden in dem Vortrag zuerst die Steinsche Methode vorgestellt und einige Grundlagen zu Mannigfaltigkeiten in Zusam- menhang mit Stochastik erklärt. Im Anschluss daran definiere ich eine geeignete Diffusion, erkläre ihre Eigenschaften, bilde sie auf R^d ab und wende die Steinsche Meth- ode der austauschbaren Paare an. Das Vorgehen entstammt dem Artikel ”Constructing exchangeable pairs by diffusion on manifold and its application” von Weitao Du, 2006.
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### 30.01.2023 16:30 Peter K. Friz (TU Berlin): Martingales and rough paths

Despite being very different in nature, martingales and rough paths have many similarities and their interplay is most fruitful. As a concrete example, I will introduce the recent notion of rough stochastic differential equations and explain its importance in filtering, pathwise control theory and option pricing under (possibly rough) stochastic volatility. (Joint work with numerous people, including Pavel Zorin-Kranich, Khoa Lê, Antoine Hocquet, Peter Bank, Christian Bayer and Luca Pelizzari.)
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For talks more than 90 days ago please have a look at the Munich Mathematical Calendar (filter: "Oberseminar Wahrscheinlichkeitstheorie").