# Peter Mühlbacher

I am a second year PhD student under supervision of Daniel Ueltschi.

**Email:** Peter "dot" Muhlbacher "at" warwick "dot" ac "dot" uk

**Website: ** peter.muehlbacher.me

**Office:** B0.15

## Publications

- 2019: "Critical temperatures of loop and Bernoulli percolation in finite dimensions"
- 2018: "Bounds on the norm of Wigner-type random matrices" with László Erdős; published in RMTA
- 2017: "Gaussian free field and Liouville quantum gravity"; Part III essay

## Research Interests

My main interests are **quantum spin systems **and in particular their **graphical representations**. One toy example is the random interchange model: Start with a graph and put a label on each vertex. At every time step one edge is chosen at random and the adjacent labels are swapped. This defines a random permutation. Various quantities of interest (to physicists) turn out to be related to the permutation cycle structure of this permutation. Questions I am interested in include

*Is there a macroscopic permutation cycle at a given time?**If there is a macroscopic permutation cycle at time t, could it be that there are only microscopic cycles at time s (s>t)?**What does the cycle structure look like?*

They depend strongly on the geometry of the underlying graph.

One might see this as an S_{N}-valued Markov process. The field of **mixing times** addresses the question how long it takes until one approaches the equilibrium distribution (typically in total variation distance) and whether there is a sharp transition. It turns out, however, that if one only looks at the cycle structure, one converges to the equilibrium, the **Poisson-Dirichlet distribution**, on much shorter time scales.

Some questions can be resolved by considering a natural coupling to a **percolation process**. Going away from the toy model to the models that are actually of interest to physicists bears a striking resemblance to going from classical percolation to the **random cluster model**.

Alternatively, the time evolution can be seen as a simple random walk on the Cayley graph of the symmetric group with generators being transpositions supported on the edges of the graph. In this case **representation theory** (essentially Fourier analysis on non-commutative spaces) provides some insights.

Yet another way to understand the random interchange model is by considering a certain **non-Markovian random walk** and determining whether it is **transient**/**recurrent**. To this end I am interested in techniques devised to prove transience for simple random walks, like **electrical networks**.

If you have ideas/advice/questions on anything above in bold (or anything else), I would love to hear about it!

## Academic CV

- 2012–2016: Bachelor, University of Vienna, Mathematics
- 2016–2017: Master, University of Cambridge, Mathematical Tripos Part III
- 2018–2022: PhD, University of Warwick