TU Budapest -- BME

talks in Fall semester 2018 -- előadásai a 2018 őszi félévben

Jacobi triple product via the exclusion process

Abstract

I will give a brief overview of very simple, hence maybe less investigated
structures in interacting particle systems: reversible product blocking
measures. These turn out to be more general than most people would think, in
particular asymmetric simple exclusion and nearest-neighbour asymmetric zero
range processes both enjoy them. But a careful look reveals that these two
are really the same process. Exploitation of this fact gives rise to the Jacobi
triple product formula - an identity previously known from number theory and
combinatorics. I will show you the main steps of deriving it from pure
probability this time, and I hope to surprise my audience as much as we got
surprised when this identity first popped up in our notebooks.

2019.01.10 Thursday, 16:15

Existence of phase transition for percolation using the Gaussian Free Field

Abstract

We prove that Bernoulli percolation on bounded degree graphs with
isoperimetric dimension d>4 undergoes a non-trivial phase transition
(in the sense that p_{c}<1). As a corollary, we obtain that the critical
point of Bernoulli percolation on infinite quasi-transitive graphs (in
particular, Cayley graphs) with super-linear growth is strictly
smaller than 1, thus answering a conjecture of Benjamini and Schramm.
The proof relies on a new technique consisting in expressing certain
functionals of the Gaussian Free Field (GFF) in terms of connectivity
probabilities for percolation model in a random environment. Then, we
integrate out the randomness in the edge-parameters using a
multi-scale decomposition of the GFF. We believe that a similar
strategy could lead to proofs of the existence of a phase transition
for various other models.

Joint work with Hugo Duminil-Copin, Subhajit Goswami, Franco Severo, Ariel Yadin.

Joint work with Hugo Duminil-Copin, Subhajit Goswami, Franco Severo, Ariel Yadin.

2018.12.13 Thursday, 16:15

Restrictions of Brownian motion

Abstract

It is classical that the zero set and the set of record times of a linear
Brownian motion have Hausdorff dimension 1/2 almost surely. Can we find a
larger random set on which a Brownian motion is monotone? Perhaps
surprisingly, the answer is negative. We outline the short proof, which is
an application of Kaufman's dimension doubling theorem for planar Brownian
motion. If time permits, we discuss related results for random walk and
fractional Brownian motion as well, and pose some open problems. This is a
joint work with Omer Angel, András Máthé, and Yuval Peres.

2018.11.29 Thursday, 16:15

Macroscopic cycles in the interchange process and the quantum Heisenberg model on the Hamming graph

Abstract

The topic of this talk will be random permutations coming from the
interchange process and its generalizaton, the quantum Heisenberg
model from statistical physics, on the 2-dimensional Hamming graph. We
prove the existence of a phase transition - in the interchange process
for sufficiently long times there exists, with high probability, a
cycle of macroscopic size, while for short times all cycles are small.
This is the first result of this kind for a graph with nontrivial
geometry, generalizing known results due to Schramm (2008). One of the
main techniques used is the cyclic time random walk. Joint work with
Radosław Adamczak and Piotr Miło¶.

unusual time / szokatlan időpont

2018.11.19 Monday, 16:15

Unusual place: Rényi Intézet, angyalkás terem.