# Munich Summer School Discrete Random Systems

## 28-30 September 2022

The summer school is a meeting for the probability community in Munich (+Augsburg) with mini-courses by international experts as well as talks by scientists from Munich. The school marks our post-covid relaunch of face-to-face scientific meetings with ample interaction with the other probabilists in the Munich area.

Scientific Program: The summer school features mini-courses by
• Mathew Penrose (University of Bath): Large components of random geometric graphs and
• Michael Drmota (TU Wien): Random Trees - An Analytic Approach.
Additionally, there will be talks by local speakers:
• Noam Berger: An improved condition for uniqueness of Doeblin measures
• Christian Kühn: Dynamics on Graph Limits
• Stefan Großkinsky: Poisson-Dirichlet asymptotics in mean-field particle systems
• Peter Müller: On the return probability of the simple random walk on supercritical Galton-Watson trees

Schedule: The program starts on Wednesday morning (departure from Munich Central Station around 9AM) and finishes on Friday late afternoon (return at Munich Central around 6PM). Weather-permitting, we plan a hike on Thursday afternoon.

Wednesday
Thursday Friday
09:04-09:56 train Munich-Schliersee 9:30-10:30 Course Drmota 09:30-10:30
Course Penrose
10:30-11:30 Course Drmota 11:00-12:00 Course Drmota 11:00-12:00 Course Drmota
12:00-13:00 Course Drmota 12:15-13:00 Talk Großkinsky 12:15-13:00 Talk Kühn
lunch lunch lunch
14:30-15:30 Course Penrose
hike 14:30-15:15 Talk Müller
15:45-16:45
Course Penrose

15:15-16:00
Talk Berger

16:34 train departure

Dinner (BBQ)

Venue: The scientific program takes place at Basislager Schliersee. Accommodation is in Schliersee and Hausham (information for registered participants later). Schliersee is easily accessible with BRB.

Mini-courses:
• Michael Drmota: Random Trees - An Analytic Approach

Many classes of random trees are directly related to corresponding "combinatorial tree classes" like binary trees, planar trees, labelled trees, increasing trees, size conditioned Galton-Watson trees etc. so that the probability distribution of random trees can be restated into counting problems on trees. The purpose of this course is to present some background on asymptotic counting methods on trees that are mainly bases on the asymptotic analysis of multivariate generating functions that satisfy finite or infinite systems of equations. This leads to various results on the limiting behavior of several tree parameters (such as heigth, profile, pattern occurences) as well as to graph limits. Similar results can be also obtained for several "tree-like" random graphs like series-parallel graphs.

Slides from Drmota's talks

• Mathew Penrose: Large components of random geometric graphs

The random geometric graph $G(n,r)$ has $n$ vertices uniformly distributed in the unit square, with edges between any two vertices distant less than $r$ from each other. This course is concerned with the asymptotic behaviour of the graph $G(n,r(n))$ (or the Poissonized version thereof) in the large-$n$ limit with a given sequence $r(n)$. We shall discuss two types of phase transition:

• If $n r(n)^2 /\log n$ tends to a constant $c$, then the graph is disconnected with high probability for $c< 1/\pi$, but connected w.h.p. for $c > 1/\pi$.
• If instead $n r(n)^2$ tends to a constant $b$, then the graph enjoys a `giant component' containing a positive proportion of the vertices, asymptotically in probability, if and only if $b$ exceeds a certain critical value.
We shall explain a number of key results and ideas in the theory of point processes and continuum percolation, which are needed to derive the above two results.

Penrose's lecture notes

Organizers: Nina Gantert, Konstantinos Panagiotou, Markus Heydenreich, Michaela Platting