# BUDAPEST-VIENNA PROBABILITY SEMINAR

### This series of events is jointly organised by the community of probabilists -- in a very wide sense -- of Vienna (IST, TUW, UW) and Budapest (BME, ELTE, RI). The informal group of organisers consists (right now) of: Mathias Beiglböck, Nathanaël Berestycki, László Erdős, Jan Maas, Miklós Rásonyi, Gábor Pete, Fabio Toninelli and Bálint Tóth.

### The events will be held quasi-regularly, with one or two meetings per semester, consisting of three 50 minutes lectures. The location of the events will alternate between the two cities Budapest and Vienna. All are welcome!

### The first, inaugural, was held in Budapest, at the Rényi Institute, on Friday, 6th of March 2020. This time, due to the circumstances, the talks will be held online via Zoom.

### Please send us an email if you want to join the online version of our BUDAPEST-VIENNA PROBABILITY SEMINAR to manuela.ehrenfried@univie.ac.at and then we will send you the link!

# 2020-22-09

# ONLINE via Zoom

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## THE TALKS:

### 14:00-14:50

### ADAM TIMAR (RI Budapest/U. of Iceland): The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generating set

**Abstract: **The uniform measure on the set of all spanning trees of a finite graph is a classical object in probability. In an infinite graph, one can take an exhaustion by finite subgraphs, with some boundary conditions, and take the limit measure. The Free Uniform Spanning Forest (FUSF) is one of the natural limits, but it is less understood than the wired version, the WUSF. If we take a finitely generated group, then several properties of WUSF and FUSF have been known to be independent of the chosen Cayley graph of the group: the average degree in WUSF and in FUSF; the number of ends in the components of the WUSF and of the FUSF; the number of trees in the WUSF. Lyons and Peres asked if this latter should also be the case for the FUSF.

In a joint work with Gábor Pete we give two different Cayley graphs of the same group such that the FUSF is connected in one of them and it has infinitely many trees in the other. Furthermore, since our example is a virtually free group, we obtained a counterexample to the general expectation, that such "tree-like" graphs would have connected FUSF. Several open questions are inspired by the results. We also present some preliminary results and conjectures on phase transition phenomena that happen if we put conductances on the edges of the underlying graph.

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### 15:00-15:50

### GIUSEPPE CANNIZZARO (University of Warwick): A new Universality Class in (1+1)-dimensions: the Brownian Castle

**Abstract: **In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively.

Starting from a "infinite-temperature" version of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. This talk is based on a joint ongoing work with Martin Hairer.

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### 16:15-17:05

**Abstract:** The most known example of an order-disorder phase transition is the Ising model that describes dependence of magnetic properties on the temperature T. This model on assignments of +1 and -1 to the sites is defined by nearest-neighbour interactions which get weaker when T increases - the correlations are uniformly positive below T_c and decay exponentially in the distance above T_c.

The loop O(n) model introduced in Physics in 1980 can be viewed as a generalisation of the Ising model where long-range interactions are added - the probability depends on the number of connected components of +1 and -1. Up to some extent, the loop O(n) model can be compared to the Fortuin-Kasteleyn (FK) percolation, though the set of parameters where it is conjectured to be conformally invariant is significantly wider.

In this talk, we will discuss recent results and open problems concerning the phase diagrams of the loop O(n) and the FK models.

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## DINNER:

We may go to dinner in Vienna after the (online) seminar. If you are interested and you are in Vienna please do inform Manuela about this by Thursday, 17th of September per email: manuela.ehrenfried@univie.ac.at

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## AND:

If you want, we can all toast together with our glasses after the talks – on Zoom, of course!

## Do you want to know what was going on in March 2020? Then you will find here the program of our PAST EVENT.

Our team in Vienna clicking and drinking a half-virtual glass of wine after our seminar on September 22, 2020.